Extended phase space thermodynamics for Lovelock black holes with non-maximally symmetric horizons
aa r X i v : . [ g r- q c ] J a n Extended phase space thermodynamics for Lovelock black holes with non-maximallysymmetric horizons
N. Farhangkhah ∗ and Z. Dayyani Department of Physics, Shiraz Branch, Islamic Azad University, Shiraz 71993, Iran Physics Department and Biruni Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran
We study thermodynamics and critical behaviors of higher-dimensional Lovelock black holes withnon-maximally symmetric horizons in the canonical ensemble of extended phase space. The effectsfrom non-constancy of the horizon of the black hole via appearing two chargelike parameters inthermodynamic quantities of third-order Lovelock black hole are investigated. We find that Ricciflat black holes with nonconstant curvature horizon show critical behavior. This is an interestingfeature that is not seen for any kind of black hole in Einstein or Lovelock gravity in the literature. Weexamine how various interesting thermodynamic phenomena such as standard first-order small-largeblack hole phase transition, a reentrant phase transition, or zeroth order phase transition happenfor Ricci flat, spherical, or hyperbolic black holes with nonconstant curvature horizon dependingon the values of Lovelock coefficient and chargelike parameters. While for a spherical black hole ofthird order Lovelock gravity with constant curvature horizon phase transition is observed only for7 ≤ d ≤
11, for our solution criticality and phase transition exist in every dimension. With a properchoice of the free parameters, a large-small-large black hole phase transition occurs. This process isaccompanied by a finite jump of the Gibbs free energy referred to as a zeroth-order phase transition.For the case κ = − PACS numbers: 04.50.-h,04.20.Jb,04.70.Bw,04.70.Dy
I. INTRODUCTION
Einstein’s theory of general relativity, is the most successful theory of gravity. At very high energies close tothe Planck scale, higher order curvature terms can no longer be neglected. Over the past years, motivated bySuperstring/M-theory [1], higher dimensional gravity has been a prevailing subject of study. The most famous theorythat generalizes general relativity in higher dimensions is Lovelock theory [2]. This theory keeps the order of the fieldequations down to second order in derivatives. The Lovelock Lagrangian consists of a sum of dimensionally extendedEuler densities, and the second-order field equations in this model give rise to the ghost-free nature of the theory.The best testing ground for any modified theory of gravity will be to search for black hole solutions. Investigatingdifferent aspects of black hole physics has raised many interests.Since it was found that black holes in classical general theory of relativity obey laws that are analogous to the lawsof thermodynamics, a lot of attentions have been attracted to study the thermodynamic properties of the black holes.First, Hawking proposed that the area of the black hole event horizon never decreases. This is analogous to the secondlaw of thermodynamics with area of event horizon playing the role of entropy in thermodynamics. After that the blackhole entropy was introduced as proportional to the black hole surface area in Planck units by Bekenstein [3]. Hawkingalso suggested the temperature for the black hole [4], and proposed that like any other hot body, the black hole radiatesin analogy to the zeroth law of thermodynamics. Thus the black hole can be considered to be a thermodynamic objectand study of black hole thermodynamics has provided interesting information about the underlying structure of thespacetime. The discovery of the famous Hawking-Page phase transition in Schwarzschild Anti de Sitter (AdS) blackholes [5] was a begin to a wide area of researches in the context of black hole thermodynamics.In thermal systems, Van der Waals equation modifies the equation of state for an ideal gas to one that approximatesthe behavior of real fluids. Through studying the thermodynamics of charged black holes, it was found that the firstorder small-large phase transition for charged black hole in AdS space is quite similar to the liquid-gas change ofphase occurring in Van der Waals fluids [6, 7]. It was showed that Q − Φ diagram of the charged black holes is similarto the P − V diagram of the van der Waals system. Researches in this regards, led to the assumption of an extendedthermodynamic phase space . In this framework, black hole thermodynamics is studied in the asymptotic AdS spacewith a negative cosmological constant Λ. The cosmological constant is represented as a pressure (Λ = − P/ π ) , andthe thermodynamically conjugate variable is the thermodynamic volume [9–12]. Also the mass of a black hole is ∗ email address: [email protected] suggested to be interpreted as the enthalpy of the spacetime instead of the internal energy [8]. It was shown that inthis extended phase space both the Smarr relation [13] and the first law of thermodynamics hold. One can also defineother thermodynamic quantities of black holes such as adiabatic compressibility, specific heat at constant pressure,or even the speed of sound [14, 15].The analysis of the P − V critical behaviors in the extended phase space has been under study extensively andgeneralized to higher dimensional charged black holes [12, 16, 17], rotating black holes [18, 19], and black holeswith Born-Infeld filed [20]. If a monotonic variation of any thermodynamic quantity results in two (or more) phasetransitions such that the final state is macroscopically similar to the initial state, the system undergoes the reentrantphase transition [19, 21]. The reentrant phase transition was found for the four-dimensional Born-Infeld-AdS blackhole spacetimes, [20], and for the black holes of third-order Lovelock gravity [28]. The situation are accompanied by adiscontinuity in the global minimum of the Gibbs free energy, referred to as a zeroth-order phase transition and seenin superfluidity and superconductivity [22].Corrections to black hole thermodynamics from higher-curvature terms in Lovelock theory have revealed interestingfeatures. In Lovelock theory the entropy is given by a complicated relationship depending on higher-curvature terms,and is no longer proportional to the area of the horizon. Also it is proposed to introduce a quantity conjugated tothe Lovelock coefficient in the first law of black hole thermodynamics and in the Smarr relation. P − V criticality has been searched for Gauss-Bonnet [23–25] and third-order Lovelock [26–29] black holes . It wasshown in [23] that P − V criticality can be observed for spherical Gauss-Bonnet black holes even when charge is absent.For third order Lovelock gravity, it is found that for κ = 1 only for dimensions 7 ≤ d ≤
11 critical points exists. Inall the cases mentioned above, for κ = 0 there is no critical point in the extended thermodynamic phase space. Allthe works mentioned above have considered black hole with maximally symmetric horizons. In Ref. [30] a novel classof black hole solution is derived with nonconstant curvature horizon in Lovelock gravity. A noteworthy change whenconsidering nonconstant curvature horizon is that new chargelike parameters appear in the metric function with theadvantage of higher curvature terms, and modify the properties of the black holes. Specially, Ricci flat solutions ofthis kind of black holes show interesting features and this motivates us to investigate the effects of non constancy ofthe horizon on the P − V criticality of such black holes.Our paper is organized as follows. We begin in Sec. II by reviewing the solutions of Lovelock gravity with non-constant curvature horizons and the extended phase space thermodynamics in Lovelock theory is discussed. In Secs.III, IV and V, specifying to black holes of 3-order Lovelock gravity, critical behavior of the black hole with nonconstanthorizon but constant sectional curvatures κ = 0 , ± II. EXTENDED THERMODYNAMICS OF NONMAXIMALLY SYMMETRIC LOVELOCK ADSBLACK HOLES
To start, we consider the physical action describing Lovelock gravity which is in the following form: I = Z M d d x √− g −
2Λ + p X p =1 α p L ( p ) ! . (1)where Λ is the cosmological constant and α p ’s are the Lovelock coupling constants with the choose of α = 1. TheEinstein term L (1) equals to R and the second order Lovelock term is L (2) = R µνγδ R µνγδ − R µν R µν + R . Also L (3) is the third order Lovelock Lagrangian which is described as L (3) = 2 R µνσκ R σκρτ R ρτµν + 8 R µνσρ R σκντ R ρτµκ + 24 R µνσκ R σκνρ R ρµ +3 RR µνσκ R σκµν + 24 R µνσκ R σµ R κν + 16 R µν R νσ R σµ − RR µν R µν + R . (2)We start with the following metric ds = − f ( r ) dt + f − ( r ) dr + r γ ij ( z ) dz i dz j , (3)which is a warped product of a 2-dimensional Riemannian submanifold M and an ( d − K ( d − . In this relation i, j go from 2 , ..., d −
1. The submanifold K ( d − with the unit metric γ ij is assumed to be anEinstein manifold with nonconstant curvature but having a constant Ricci scalar being e R = κ ( d − d − , (4)with κ being the sectional curvature. For the tensor components of the submanifold K ( d − a tilde is used. The Ricciand Riemann tensors of the Einstein manifold are e R ij = κ ( d − γ ij , (5) e R ij kl = e C ij kl + κ ( δ ik δ jl − δ il δ jk ) , (6)where e C klij is the Weyl tensor of K ( d − .Choosing p = 3 in the field equation, for the metric (3) to be a solution of field equations in third order Lovelocktheory in vacuum, it would suffice that the Weyl tensor of the horizon satisfies the following constraints X kln e C kinl e C nlkj = 1 d δ ij X kmpq e C kmpq e C pqkm ≡ η δ ij , (7) X klnmp e C nmpk e C klni e C pjml − e C pmni e C jnkl e C klpm )= 2 d δ ij X klmpqr (cid:16) e C qmpk e C klqr e C prml − e C pmqr e C rqkl e C klpm (cid:17) ≡ η δ ij . (8)The first constraint was originally introduced by Dotti and Gleiser in [30] and is due to the Gauss-Bonnet term, andthe second one which is dictated by the third order Lovelock term, is obtained in [31]. These two new chargelikeparameters appear in the metric function with the advantage of higher curvature terms, and modify the properties ofthe black holes.Considering the case α = α ( d − d −
4) (9) α = α (cid:0) n − (cid:1) , (10)the metric function f ( r ) is given by [31] f ( r ) = κ + r α (cid:26) (cid:16) j ( r ) ± p γ ( r ) + j ( r ) (cid:17) / − γ ( r ) / (cid:16) j ( r ) ± p γ ( r ) + j ( r ) (cid:17) − / (cid:27) ,j ( r ) = −
12 + 3 α (cid:18) − d − d − − mr d − + α ˆ η r (cid:19) ,γ ( r ) = (cid:18) α ˆ η r (cid:19) , (11)where we define ˆ η = ( d − η ( d − and ˆ η = ( d − η ( d − for simplicity. Note that α and ˆ η are positive parameters, while ˆ η can be positive or negative relating to the metric of the spacetime. We should also mention that in order to have theeffects of non-constancy of the curvature of the horizon in third order Lovelock gravity, d should be larger than seven,since the constants ˆ η and ˆ η are evaluating on the ( d − P = − Λ8 π , (12) V = ( ∂M∂P ) S,α = Σ d − r d − h d − , (13)We obtain the parameter M in terms of the horizon radius r h by solving f ( r ) = 0 as below M = ( d − d − π [ 16 πP ( d − d − r d − h + κr d − h + α ( κ + ˆ η ] r d − h + α κ + 3ˆ η κ + ˆ η ) r d − h ] (14)which is interpreted as enthalpy rather than the internal energy of the gravitational system. Σ d − denotes the volumeof the ( d − K ( d − . The Hawking temperature of such black holes, related with the surfacegravity on the horizon r = r h is given by [31] T = πP ( d − r h + ( d − κr h + ( d − η + κ ) αr h + ( d − α (ˆ η + 3 κ ˆ η + κ )4 πr h [ r h + 2 καr h + α (ˆ η + κ )] , (15)and entropy can be derived by making use of the Wald prescription as S = ( d − d − r d − h d −
2) + 2 καr h ( d −
4) + α (ˆ η + κ ) r h ( d −
6) ] . (16)The first law, in the extended phase space, yields dM = T dS + V dP + A dα (17)where A denote the quantities conjugated to the Lovelock coefficient and is calculated as below A = ( ∂M∂α ) S,P = ( d − d − r d − h π { r ( κ + b η ) + 2 α ( κ + 3 κ b η + b η ) − ( κ ( d − r + α ( κ + b η )( d − ) r + 2 καr + α ( κ + b η ) [ 48 πr Pd − κ ( d − r + 3( d − α ( κ + b η ) r + ( d − α ( κ + 3 κ b η + b η )] } . (18)These thermodynamical quantities satisfy the generalized Smarr relation in the extended phase space M = d − d − T S − d − V P + 2 d − A α (19)One can rearrange Eq. (15) to get thermodynamic equation of the state for the black hole in the following form, P = Tv − κ ( d − π ( d − v + 32 καT ( d − v − α ( d − η + κ ) π ( d − v + 256 α T (ˆ η + κ )( d − v − α ( d − κ + 3 κ ˆ η + ˆ η )3 π ( d − v , (20)in which we have introduced the parameter v = 4 r h ( d −
2) (21)as an effective specific volume.If we consider the Van der Waals equation given as P = Tv − b − av , (22)and make use of the series expansion (1 − bv ) − = X n =0 ( bv ) n , (23)it is well seen that if we keep the higher order terms in the Taylor series expansion, the Van der Waals equation is incorrespondence with the equation of state (20) including the terms which appear from Lovelock gravity.The critical point occurs when P = P ( v ) has an inflection point, i.e., ∂P∂v = 0 , ∂ P∂v = 0 (24)and ∂ P∂v changes signs around each of the solution. One of the best ways to investigate the critical behavior and phasetransition of the system is to plot the isotherm diagrams and compare with Van der Waals liquid-gas system. In whatfollows we shall investigate the P − v criticality of the black hole with nonconstant horizon but constant sectionalcurvatures κ = 0 , ± . (a) α = 1, ˆ η = 1, ˆ η = − d = 11 (b) α = 1, ˆ η = 2, ˆ η = − d = 8 FIG. 1: P − v diagram of Lovelock black holes with κ = 0 . III. CRITICAL BEHAVIOR OF LOVELOCK RICCI FLAT BLACK HOLES WITH κ = 0 For κ = 0, the equation of state can be written as P = Tv − α ( d − η π ( d − v + 256 α ˆ η T ( d − v − α ( d − η π ( d − v , (25)To obtain the critical points, if exist, we should solve Eqs. (24) which could be simplified as x + qx + s = 0 , x = v (26)with the parameters q and s given by q = −
53 ˆ η B − C , s = 35( d − η d − B + 2 C B = 16 α ( d − , C = 40ˆ η ( d − α η ( d − ( d − . (27)As we mentioned before, ˆ η is a positive parameter but ˆ η can take an arbitrary positive or negative value. It iswell known that the multiplication of three roots of Eq. (26) is proportional to − s which can be shown through astraightforward calculation to be proportional to − ˆ η . Thus for negative values of ˆ η , the multiplication of three rootsof Eq. (26) is positive and thus there exists at least one positive real root for this equation. This fact leads to theexistence of at least one real root for Eqs. (24). One should note that in the expression for P which is given by therelation (25), the last term is dominant, as v → , which is positive for negative values of ˆ η . The isotherm diagrams P − v for a Ricci flat black hole are displayed in Fig. 1 for two values of d . The plots obviously show a first orderphase transition in the system for T < T c which is really similar to the Van der Waals liquid-gas system. As it isseen, for a fixed temperature lower than the critical one, in the small radius region and large one the compressioncoefficient is positive, which shows stable phases. Between them there is an unstable phase. therefore a small/largeblack hole phase transition occurs.It is worthwhile to emphasis that such a phase transition is never seen for ˆ η = ˆ η = 0. As is well known any planarblack holes with constant curvature horizon of Einstein or higher-order Lovelock gravity in an arbitrary number ofspacetime dimensions in vacuum or even in the existence of Maxwell, Born-Infeld, or dilaton fields do not admit criticalbehavior. This interesting behavior is due to the existence of ˆ η and ˆ η which appear as a result of the nonconstancyof the horizon and makes drastic changes to the equation of state in the case κ = 0 . Also there is no criticality forˆ η = 0 and ˆ η = 0. This reveals the effect of higher-curvature terms in third-order Lovelock gravity, which cause novelchanges in the properties of the spacetime. (a) α = 1, ˆ η = 2, ˆ η = 5 and d = 8 (b) α = 1, ˆ η = 2, ˆ η = 2 and d = 8 FIG. 2: P − v diagram of Lovelock black holes with positive ˆ η for κ = 0 The solutions to Eq. (26) could be written as v c = √ x = [ s − s r ( q + ( s + s − s − r ( q + ( s ] , (28) T c = ( d − η d − π ˆ η v c + 380 ( d − d − απ v c . (29)For positive values of ˆ η , the relation (28) makes a limitation on ˆ η that depends on α , ˆ η and number of dimensions d as ˆ η < √ d − η / d − . (30)For positive values of ˆ η , satisfying the above constraint, Eq. (26), has at least one real root introduced as v c . Towitness the P − v criticality behavior we plot the P − v diagram in Fig. 2. P − v diagrams in diverse dimensions arethe same and so without loss of generality we present them for d = 8. Fig. 2(b) show that for some values of ˆ η , inevery dimension, there are two critical points, one with negative (unphysical) and the other with positive pressure.The isothermal plots in this case are quite similar with the P − v diagram of Born-Infeld-AdS black holes [20]. As ˆ η is increased up to the limiting value obtained in the relation (30), both values of critical pressure become positive asis seen in 2(a). A. Gibbs free energy
One of the most important items that helps us to determine phase transition of a system refers to study itsthermodynamic potential. Gibbs free energy generally is computed from the Euclidean action with appropriateboundary term [32] while the lowest Gibbs free energy is associated with global stable state. In the canonical ensembleand extended phase space, thermodynamic potential closely associates with the Gibbs free energy G = M − T S . As iswell known, to have a physical behavior, the second order derivative of Gibbs energy with respect to the temperatureshould be negative to have a positive heat capacity. Zeroth order phase transition occurs in the system when Gibbsenergy is discontinuous. This behavior was formerly observed in superfluidity and superconductivity [33]. Anydiscontinuity in fist (second) order derivatives of Gibbs energy leads to a first (second) order phase transition in the (a) α = 1, ˆ η = 1, ˆ η = − d = 11 (b) α = 1, ˆ η = 2, ˆ η = − d = 8 FIG. 3: Gibbs diagrams of Lovelock black holes for ˆ η < κ = 0 system. We calculate the Gibbs free energy of the black hole to elaborate the phase transition of the system as below, G = G ( P, T ) = − P r d − h [5 α ( d − η + ( d − r h ]( d − d − d −
1) ( α η + r h )+ η r d − h [ α ( d − (cid:0) d − d + 2 (cid:1) η + 5 α ( d − (cid:0) d − d + 2 (cid:1) r h )48 π ( d − d − d −
1) ( α η + r h )+ r d − h (cid:0) α ( d − (cid:0) d − d + 2 (cid:1) η r h − α ( d − (cid:0) d − d + 2 (cid:1) η r h (cid:1) π ( d − d − d −
1) ( α η + r h ) (31)where r h should be understood as a function of pressure and temperature via the equation of state. The Gibbs freeenergy corresponding to Fig. 1 is depicted in Fig. 3. One can note that for negative η , the Gibbs energy has asmooth behavior as a function of T , for P > P c whereas for P < P c it exhibits the small/large black hole first orderphase transition and a usual swallowtail shape as we expect which is the characteristic of van der Waals fluid.The corresponding Gibbs diagram to Fig. 2 with two critical points is displayed in Fig. 4. In Fig. 4(a) when P ≤ P c the lower (upper) branch is thermodynamically stable(unstable). There is only one physical branch andGibbs energy shows no phase transition in the system. However, a first order phase transition may happen in therange of P c < P < P c as shown with solid red line in Fig. 4(a). In Fig. 4(b) we can see a first order phasetransition similar to Van der Waals fluid in the range 0 < P < P c and thus the second critical point (with P = P c )is physical. The critical point with negative pressure does not globally minimize the Gibbs energy and hence isnot physical. Although the equation of state leads to one or two critical point in this case, investigating the Gibbsdiagrams represents only one physical critical point. B. Critical exponents
Critical exponent characterizes the behavior of physical quantities in the close vicinity of the critical point. Weproceed to calculate the critical exponents α ′ , β ′ , γ ′ and δ ′ for the phase transition of a d -dimensional Lovelock blackhole. In order to calculate the first critical exponent α ′ , we consider the entropy S given by Eq. (16) as a function of T and v . Making use of Eq.(21) we have S = S ( T, v ) = 4 − d ( d − d − v d − + α − d ( d − d − v d − ˆ η d − (a) α = 1, ˆ η = 2, ˆ η = 5 and d = 8. One has twocritical points with positive pressure while only the onewith P = P c corresponds to the first order phasetransition between small and large black holes. Theother does not globally minimize the Gibbs free energyand hence is unphysical. (b) α = 1, ˆ η = 2, ˆ η = 2 and d = 8. There are twocritical points with positive and negative pressure. Asone expects only T = T c with positive pressurecorresponds to the first order phase transition. Theother critical point ( T = T c ) is unphysical FIG. 4: Gibbs diagrams of Lovelock black holes for ˆ η > κ = 0. The curves are shifted and rescaled for more clarity. It is clear that entropy does not depend on the temperature in this relation and hence C V = 0. This indicates thatrelative critical exponent will be zero C V ∝ (cid:18) TT c − (cid:19) α ′ ⇒ α ′ = 0 . (33)To obtain the other exponents, we define the reduced thermodynamic variables as p ≡ PP c , ν ≡ vv c , τ ≡ TT c , and expansion parameters as t = τ − , ω = ν − vv c − . (34)Then we can make Taylor expansion for the equation of state Eq.(25) as p = 1 + At − Btω − Cω + O (cid:0) tω , ω (cid:1) , (35)where A , B and C are constants depending on d , α , ˆ η and ˆ η .Denoting the volume of small and large black holes by ω s and ω l , respectively, differentiating Eq. (35) with respectto ω at a fixed t <
0, and applying the Maxwell’s equal area law [17] one obtains p = 1 + At − Btω l − Cω l = 1 + At − Btω s − Cω s − P c Z ω s ω l ω (cid:0) Bt + 3 Cω (cid:1) dω, (36)which leads to the unique non-trivial solution ω l = − ω s = r − BtC . (37)Thus, the exponent β ′ , which describes the behaviour of the order parameter η = v c ( ω l − ω s ) on a given isotherm,may be calculated through the use of Eq. (37) as: η = 2 v c ω l = 2 r − BtC = ⇒ β ′ = 12 . (38)To calculate the exponent γ ′ , we may determine the behavior of the isothermal compressibility near the critical point κ T = − V ∂V∂P (cid:12)(cid:12)(cid:12) T ∝ | t | − γ ′ . Since dv/dω = v c , the isothermal compressibility near the critical point reduces to κ T = − V ∂V∂P (cid:12)(cid:12)(cid:12) T ∝ V c BP c t , (39)which shows that γ ′ = 1. Finally the ‘shape‘ of the critical isotherm t = 0 is given by (35) p − − Cω , (40)which indicates that δ ′ = 3.The critical exponents associated with this type of Lovelock black holes are independent of metric parameters andthe dimension of the spacetime. This is consistent with the results of mean field theory that believe the criticalexponents are universal and do not depend on the details of the physical system. IV. CRITICAL BEHAVIOR OF LOVELOCK SPHERICAL BLACK HOLES WITH κ = 1 When the topology of the black hole horizon is spherical the equation of state is in the form P = Tv − d − π ( d − v + 32 αT ( d − v − α ( d −
5) [ˆ η + 1] π ( d − v + 256 α [ˆ η + 1] T ( d − v − α ( d −
7) [3ˆ η + ˆ η + 1]3 π ( d − v (41)In different types of black holes, first order phase transition occurs for black holes which have spherically symmetrichorizon. Therefor it is important to investigate this case ( κ = 1) and compare our results with the other types ofblack holes.It is shown in [26], that for third order Lovelock black holes with κ = 1 and constant curvature horizon there existtwo critical points for 8 ≤ d ≤
11 and no critical point exists for d > v + bv + cv + dv + e = 0 (42)with b = 96 ˆ η ( d − − d − c = 256 5( d − η + 12( d − η + 2( d − d − d = 8192 9( d − η − d − η + (17 d − η + 2(2 d − d − e = 327680( d −
7) [(ˆ η + 1)ˆ η + 1] + 3ˆ η + 4ˆ η d − . (43)A well-known calculation yields∆ = b c d − b c e − b d + 18 b cde − b e − c d +16 c e + 18 bcd − bc de − b d e + 144 b ce − d + 144 cd e − c e − bde + 256 e . (44)0 (a) d = 8 , α = 1, ˆ η = 1 and ˆ η = −
5. (b) α = 1, ˆ η = 1 and ˆ η = 0 . d = 8 , α = 1, ˆ η = 1 and ˆ η = − . FIG. 5: P − v Diagram of Lovelock black holes with spherical horizon in d = 8 which shows a Van der Waals behavior (a), andthe possibility of reentrant phase transition (b) and (c). For the equation (42) to have solution, ∆ should be positive, which again makes a constraint on the parameter ˆ η relating to ˆ η , α and d. On the other hand, a look at the relation (41) reveals that the dominant term is the lastone, which is positive for ˆ η ≤ − (3ˆ η + 1). If ˆ η is chosen in such a way that this inequality and also ∆ > ∞ as v → η satisfying ˆ η > − (3ˆ η + 1), and the constraint ∆ >
0, the pressure tends to −∞ as v → d . TABLE I: Critical values in different dimensions for κ = 1 d α η η v c T c P c v c T c P c . . . . . . . − . . . − . . . . − . . . − . . . . . . . − . . − . .
050 0 . . − . . . − . . . . . . − . . − . . . . − . . . The corresponding P − v diagrams for diverse choices of η are depicted in Fig 5. The diagrams are the same in anydimension d . So without loss of generality we plot them for d = 8. The interesting point that one should note is thatdespite of the case for the black holes with constant curvature horizon, there exists criticality for d ≥
11. As one cansee, for some choices of ˆ η two critical points are present, one with negative (unphysical) and the other with positivepressure (5(c)). But one can find values for free parameters for which two critical points with positive pressure existas is seen in Fig. 5(b). This behavior is reminiscent of the interesting reentrant phase transition. So we go through1 (a)The lower (upper) branch isthermodinamicallystable(unstable). There is onlyone physical branch and nophase transition occurs in thesystem. (b)A first order phasetransition occurs in T anda zeroth order in T . Inaddition a finite jump inGibbs diagram leads to azeroth order phasetransition between smalland large (intermediate)black hole. (c)The first order phasetransition finishes in thecritical point P = P c . For P > P c , we see only onestable (lower) branc withno phase transition. FIG. 6: Gibbs diagram with d = 8, α = 1, ˆ η = 1 and ˆ η = 0 . the Gibbs plot for these cases. The Gibbs free energy obeys the following thermodynamic relation for any value of κG = P r d − d − d − kr d − π + α ( d − r d − (cid:0) ˆ η + k (cid:1) π + α ( d − r d − (cid:0) η k + ˆ η + k (cid:1) π − r d − (cid:0) α ( d − d − (cid:0) ˆ η + k (cid:1) + 2 α ( d − d − kr + ( d − d − r (cid:1) π ( d − d − d − (cid:16) α ˆ η + ( αk + r ) (cid:17) × (cid:8) α ( d − d − (cid:0) η k + ˆ η + k (cid:1) + 3 α ( d − d − r (cid:0) ˆ η + k (cid:1) + 3( d − d − kr + 48 πP r (cid:9) (45)The behavior of Gibbs free energy for a positive value of ˆ η is depicted in Fig. 6. Two positive critical pressuresare T c and T c . Looking at Fig. 6(a) we observe that between P c = 0 .
013 and P = 0 . < P c , the lower branch isglobally stable and therefore no phase transition could happen.By increasing the pressure, zeroth and first order phase transition take place between P = 0 .
053 and P = 0 . . < P ≤ P c = 0 .
063 the zeroth order phase transition disappearsand only first order phase transition could bee seen. Finally, the first order phase transition finishes at second criticalpoint P = P c = 0 .
063 as is seen in Fig. 6(c). For
P > P c , the lower branch is the unique globally stable branch andwe do not expect any phase transition.As one can see in Fig. 5(c), there are two critical points with positive and negative pressure. Here we would like tostudy the associated Gibbs free energy in Fig. 7. In the region 0 < P < . P = 0 .
04 in Fig.7(b). A first order phase transition occurs at T , where the Gibbs diagram is continuous but itsderivative is not. Also, there is a finite gap in Gibbs diagram in T and so a zeroth order phase transition occursin this point. Fig.7(b) represents a reentrant phast transition between LBH/ SBH/ LBH . More increasing of thepressure leads to the disappearance of zeroth order phase transition. For example in P = 0 . P c = 0 . P = 0 . < P c , one sees a first order phase transition and for P = 0 . > P c , no phase transition2 (a)A zeroth order phase transitionoccurs in T . The only stablebranch is lower one which isshowed by solid blue line. Thediagram showes a finite jumpbetween stable and unstablebranch and physically can notexist. (b)The first order phasetransition occurs in T and a zeroth order in T .A reentrant phasetransition occurs betweenLBH /SBH /LBH. (c)The first order phasetransition finished in thecritical point P = P c .There is only one stable(lower) branch For P > P c and so no phase transitionoccurs. FIG. 7: Gibbs diagram with d = 8, α = 1, ˆ η = 1 and ˆ η = − . T - r h diagram and the corresponding G − T curve (inset) of Lovelock black hole.As temperature decreases the black hole follows direction of arrows. The first and zeroth order phase transition are identifiedby black dotted and green dot-dashed curve, respectively. A large/small/large(intermediate) corresponds to a reentrant phasetransition. The positive (negative) sign of heat capacity is displayed by the blue solid (dash red) line. We have changed thesize of diagram and shifted for more clarity but this diagram can represent the behavior of Figs. 6(b) and 7(b). happens in the system. While T = T c is the critical point for which second order phase transition occurs, because ofthe existence of only one stable branch for P > P c , no phase transition may occur.Now, we go through the study of the behavior of isobar T − r h diagram and the corresponding G − T curve (inset)for Lovelock black hole with nonconstant curvature horizon. As it is illustrated in Fig. 8 by decreasing the radius ofhorizon in the G − T plane (the inset in Fig. 8), black hole follows the lower solid blue branch until it reaches T andchanges the direction to switch to left solid blue curve in G − T plane with a first order LBH/SBH phase transition.This is identified by black dotted line in T − r h plane. In the case of more decreasing of r h , the system experienceszeroth order phase transition between small and large black hole at T by a finite jump ( inset in Fig. 8) which isshown by dot-dashed green line in T − r h diagram in Fig. 8. Eventually, the black hole tracks the blue solid line tothe end. We should mention that we have shifted the diagram for more clarity but this diagram can represent thebehavior of Figs. 6(b) and 7(b).The same as what we observed in the case of Ricci flat black holes with κ = 0 in Sec. III, the entropy does not3depend on the temperature and so the first exponent α ′ equals zero for κ = 1. Following the approach discussed inSec.(III B, we can write the reduced equation of state as p = 1 + At − Btω − Cω + O (cid:0) tω , ω (cid:1) . (46)Therefore, it is easy to show that the critical exponents read β ′ = 12 , γ ′ = 1 , δ ′ = 3 (47) V. CRITICAL BEHAVIOR OF LOVELOCK HYPERBOLIC BLACK HOLES WITH κ = − When the topology of the black hole horizon is hyperbolic, the equation of state reads P = Tv + d − π ( d − v − αT ( d − v − α ( d − η + 1) π ( d − v + 256 α (ˆ η + 1) T ( d − v − α ( d − − η + ˆ η − π ( d − v . (48)In addition, the Gibbs free energy of the black hole can be calculated from Eq.(45) by substituting κ = − κ = − κ = 1 which we discussed in Sec. IV, the equation of state, (Eqs. 24) leads to a polynomialof degree 4 and numeric calculations show that depending on the values of the parameters α , ˆ η , and ˆ η there mayexist one or two physical critical points. In addition, we find some values for the free parameters that results in threecritical points, with two of them having positive pressure and one with negative pressure. We could not find any casewith three positive critical pressure. TABLE II: Critical values in different dimensions for κ = − d α ˆ η ˆ η T c P c T c P c T c P c In Fig. 9, we have only one critical point for chosen parameters d , ˆ η , ˆ η and α . This case is exactly similar to theVan der Waals phase transition with the same isotherm curves. The swallowtail shapes of Gibbs diagras verify thefirst order phase transition too. In Fig. 10, we choose the parameters so that we can observe two physical criticalpoints. The relevant Gibbs energy in Fig. 10(b) shows two critical curves associated with P c and P c . There is nophase transition for P c < P < P c . Also the Gibbs energy and its derivatives with respect to the temperature arecontinuous. We can see first order phase transition for P < P c or P > P c .The interesting case is the case with three critical points for Lovelock black holes with κ = − P − v plot is depicted in Fig. 11. Two critical points withpositive pressure are shown in Fig. 11(a) and the corresponding Gibbs free energy is depicted in Fig. 11(b). Thethird critical point is far from the others and so we bring it in a separate diagram. As it is seen in Fig. 11(c), thethird critical point has negative pressure. Fig. 11(d) represents the Gibbs free energy corresponding to this point. Itis worth to note that for P ≤ P c the Gibbs energy is completely unphysical with negative compressibility while for P > P c there exists some physical part in Gibbs diagram. We should emphasis that the curves in Gibbs diagramsare rescaled and shifted for more clarity.Following the approach in Sec. III B and using reduced thermodynamic variables and Taylor expansion for theequation of state, Eq.(48), we obtain critical exponents as α ′ = 0 , β ′ = 12 , γ ′ = 1 , δ ′ = 3 (49)4 (a) P − v diagram (b)Gibbs diagram. FIG. 9: Critical points: κ = − α = 1, ˆ η = 10, ˆ η = − . d = 10. There exists a first order phase transition with onephysical critical point similar to the Van der Waals system. (a) P − v diagram (b)Gibbs diagram FIG. 10: Critical points: κ = − α = 1, ˆ η = 0 . η = − . d = 8. There are two physical critical point in the abovediagrams. We have changed the scale of Gibbs diagrams in the way that two critical curves are visible in one diagram. which is consistent with Van der Waals exponents and the results of mean field theory. It is worthwhile to emphasisethat in a special case with ˆ η = ˆ η = 0, a peculiar isolated critical point emerges for hyperbolic black holes and ischaracterized by non-standard critical exponents, which is discussed in details in Ref. [28]. VI. CONCLUDING REMARKS
In this study we presented some thermodynamic behaviors of black holes of a more general class of solutionspossessing non-constant curvature horizons. The horizon space of these kinds of black holes is nonmaximally symmetricEinstein space. Nontrivial Weyl tensor of such exotic horizons is exposed to the bulk dynamics through the higher-order Lovelock term. Investigating the P − v criticality behavior of such black holes of Lovelock gravity in the extendedphase space led to interesting and qualitatively new behaviors. By introducing the conjugate quantity to Lovelockparameter α , We showed that the first law of thermodynamics and the Smarr formula hold. By considering thethermodynamics of these kinds of black holes with nonmaximally symmetric horizons in cubic Lovelock gravity, wehave found some particularly novel and interesting results. As it is well-known no criticality has been found for allknown types of Ricci flat black holes in Einstein or Lovelock theories of gravity. Thus we went through Ricci flatblack holes with nonconstant curvature horizon and we found that there exists criticality in every dimension d > κ = 0. We obtained the exact solutions by solving the cubic equation and showed that relatingto the values of chargelike parameters appearing in the metric function, Van der Waals-like behavior and first order5 (a) P − v diagram for two of the threecritical points (b)Gibbs diagram for two of the threecritical points.(c) P − v diagram for the third criticalpoint. (d)Gibbs diagram for the third criticalpoint. FIG. 11: Critical points: κ = − α = 1, ˆ η = 0 .
7, ˆ η = 3 . d = 9 phase transition may happen. For some values of ˆ η , which is a chargelike parameter that is inserted in the metricfunction due to the appearance of third-order curvature terms, two critical point emerge. We have also computed thecritical exponents of the phase transition and found that in the canonical ensemble the thermodynamic exponentscoincide with those of the Van der Waals fluid.For the black holes with spherical and constant curvature horizons critical points do not exist for d >
11. For oursolutions with non constant curvature horizon we carried out the study numerically and found that one or two criticalpoints exist in every dimension even dimensions higher than 11 with the proper choices of the parameters. We saw howthe value of the parameters that are being emerged in the solutions as a result of the nonconstancy of the curvatureof the horizon, affect the types of phase transition. To disclose the phase structure of the solutions and classify theirtypes, we studied the Gibbs free energy. For κ = 1 two different behaviors have been found. For some values of thefree parameters, a first order phase transition occurs between small and large black holes which is accompanied bya discontinuity in the slop of Gibbs free energy at transition point. We showed that if the parameters adopt someproper values, a large-small-large black hole transition would happen. This process was shown to be accompaniedby a finite jump of the Gibbs free energy referred to as the zeroth-order phase transition. While for the black holeswith hyperbolic horizon in different theories of general relativity, phase transition is rarely seen, we showed that forour solution in the case κ = −
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