Universal criticality of thermodynamic curvatures for charged AdS black holes
UUniversal criticality of thermodynamic curvatures forcharged AdS black holes
Seyed Ali Hosseini Mansoori, a Morteza Rafiee a and Shao-Wen Wei b a Faculty of Physics, Shahrood University of Technology, P.O. Box 3619995161, Shahrood, Iran b Institute of Theoretical and Physics Research Center of Gravitation, Lanzhou University,Lanzhou 730000, People’s Republic of China
E-mail: [email protected], [email protected],[email protected]
Abstract:
In this paper, we analytically study the critical exponents and universal ampli-tudes of the thermodynamic curvatures such as the intrinsic and extrinsic curvature at thecritical point of the small-large black hole phase transition for the charged AdS black holes. Atthe critical point, it is found that the normalized intrinsic curvature R N and extrinsic curva-ture K N has critical exponents 2 and 1, respectively. Based on them, the universal amplitudes R N t and K N t are calculated with the temperature parameter t = T /T c − where T c thecritical value of the temperature. Near the critical point, we find that the critical amplitude of R N t and K N t is − when t → + , whereas R N t ≈ − and K N t ≈ − in the limit t → − .These results not only hold for the four dimensional charged AdS black hole, but also for thehigher dimensional cases. Therefore, such universal properties will cast new insight into thethermodynamic geometries and black hole phase transitions. a r X i v : . [ g r- q c ] J u l ontents Since the establishment of the four laws of black hole thermodynamics [1, 2], the study of thephase transition has been one of the increasingly active areas among the black hole physics.It also provides us an intriguing approach to peek into the microstructures of black holes.Microscopic interaction is also expected to be uncovered based on the statistical physics.Recently, the cosmological constant was interpreted as the thermodynamic pressure andits conjugate quantity as a thermodynamic volume of a black hole. This gives rise to newinvestigations of the black hole phase transition in the extended phase space [4–6]. Besidesthe Hawking-Page phase transition that takes place between thermal radiation and stablelarge black holes, the small-large black hole phase transition of a charged AdS black holewas completed, reminiscent of the liquid-gas phase transition of the van der Waals (VdW)fluid [6]. The phase transition starts at zero temperature. Then with increasing temperature,the phase transition point extends and ends at a critical point, where the first order phasetransition becomes a second order one. At the critical point, it was found that it shares thesame critical exponents with the VdW fluid. Subsequently, this study was generalized to otherblack hole backgrounds, and more phase transitions and phase structures were revealed, suchas the reentrant phase transition, triple point, λ -line phase transition [6–22], or see [23, 24]for a recent review.On the other hand, Riemannian geometry provides us with a useful tool to study somethermodynamic aspects like phase transitions and critical behaviors [25]. For instance, Wein-hold proposed in the thermodynamic equilibrium space a Riemannian metric defined by aHessian of the internal energy function [26]. By applying the fluctuation theory to equilibriumstates, Ruppeiner [25, 27] also introduced a different metric structure which is constructed by– 1 –he second derivatives of the entropy function. It turns out that Weinhold metric is confor-mally related to Ruppeiner metric with the temperature T as the conformal factor.It is generally believed that a phase transition is a competition between the microscopicinteraction and thermal motion of micro degrees of freedom of a black hole. Thus combiningthe phase transition and the thermodynamic geometry, we can investigate the microstructuresof black holes. This idea was first implemented in Ref. [28], where the interaction betweentwo black hole molecules was uncovered by constructing the Ruppeiner geometry.As suggested in Ref. [25], the thermodynamic scalar curvature (intrinsic curvature) scaleslike the correlation length of the system and goes to infinity at the critical point. However,in some contradictory examples [28–30], the scalar curvature of the Ruppeiner geometry doesnot diverge exactly at the critical point. To tackle this problem, a new formulation of theRuppeiner geometry, constructed from considerations about the thermodynamic potentialsthat are related to the energy function (instead of the entropy) by Legendre transformations,proposed in Refs. [31–33]. This new formalism of thermodynamic geometry (NTG) providesus with a one-to-one correspondence between crtical points where phase transitions occurand curvature singularities [33]. It is worth mentioning that one of the authors of this paperobtained independently the same metric in Ref. [34].Using such a formalism and associated features, one can find that both attractive andrepulsive interactions can dominate between these micromolecules [34, 35]. This is quitedifferent from the VdW fluid, where only an attractive interaction dominates. The resultuncovers a significant microscopic property of charged AdS black holes. In particular, thecritical behaviors were observed [34, 35], based on which the correlation length can be wellconstructed. This study was also applied to the five dimensional neutral Gauss-Bonnet AdSblack hole [36]. The interesting result suggests that the interactions can keep unchanged asthe system undergoes a phase transition, whereas its microstructures have a huge change.Other studies can also be found in [37–44].Among the study of the phase transition, critical phenomena including critical exponentsand scaling laws can reveal universal properties near a critical point for the system. So it issignificant to explore critical behavior for the thermodynamic quantities. As shown in Refs.[34, 35], it was found that the intrinsic curvature goes to negative infinity at the critical point.Further, we observed a critical exponent 2 and a universal critical amplitude -1/8. Thesequantities disclose the critical behavior of the new formalism of the Ruppeiner geometry.Extrinsic curvature geometry is also an interesting approach in the thermodynamic statespace to explore the phase transition. It has been shown in [45] that the extrinsic curvatureof a certain hypersurface in the thermodynamic space contains useful information about thelocation of the second-order critical point and stability of a system. It means that the extrinsiccurvature shares the same divergent points and signs with specific heat on such hypersurfaces.This idea has also been extended to the geometrothermodynamics (GTD) geometry [46]. Inthis paper, we aim to examine the critical exponents and universal amplitudes for the intrinsiccurvature of the whole thermodynamic manifold and the extrinsic curvature on isothermhypersurfaces immersed in the thermodynamic manifold.– 2 –his paper is organized as follows. In the next Section, we employ the new formalismof the thermodynamic geometry (NTG) to define the thermodynamic curvatures. We thenrecall some basic facts about criticality behaviors of the VdW fluid in Section 3 and apply ourapproach to such a fluid for deriving critical exponents and amplitudes for the thermodynamiccurvatures near the critical point. In Sections 4 and 5, we examine such criticality behaviorsof the thermodynamic curvatures for the AdS black holes in arbitrary dimensions. Finally,Section 6 is devoted to discussions of our results. In this section, we try to define the thermodynamic curvatures in simple form which canbe useful for further consideration in next Sections. For this purpose, we consider the newformalism of the thermodynamic geometry (NTG) [33] which is defined by dl NT G = 1 T (cid:18) η ji ∂ Ξ ∂X j ∂X l dX i dX l (cid:19) (2.1)where η ji = diag( − , , ..., and Ξ is the thermodynamic potential and X i can be intensive(extensive) variables. Note that this formalism is able to explain a one-to-one correspon-dence between phase transitions and curvature singularities. For example, in two dimensionalthermodynamic manifold, taking free energy as the thermodynamic potential, Ξ = F , and X i = ( T, x ) , where x can be any one of the volume V , charge Q , angular momentum J ,or other extensive quantities, the curvature singularities correspond precisely to the phasetransitions of the heat capacity at y constant, C y . For this case, the line element is given as dl NT G = 1 T (cid:16) − ∂ F∂T dT + ∂ F∂x dx (cid:17) (2.2)which is in agreement with that obtained in Refs. [34, 35] . Using differential form of the freeenergy, dF = − SdT + ydx , one can rewrite the above metric elements as follows, dl NT G = 1 T (cid:16) ∂S∂T (cid:17) x dT + 1 T (cid:16) ∂y∂x (cid:17) T dx = C x T dT + 1 T (cid:16) ∂y∂x (cid:17) T dx , (2.3)in which C x = T (cid:16) ∂S∂T (cid:17) x is the heat capacity at constant x . In this paper, we are interestedin applying this metric for charged AdS black holes in the extended phase space, where thecosmological constant is treated as thermodynamic pressure and its conjugate quantity asthermodynamic volume of the balck hole. Indeed, the critical behaviors of charged AdS blackholes are very similar to the VdW fluid. Although, these cases have vanishing heat capacityat constant volume, i.e., C V = 0 [35], in comparison with constant heat capacity of the VdWfluid, C V = k B , specific heat capacity of the charged AdS black hole can be treated as– 3 –he limit k B → + of the VdW fluid [35]. Thus, it is convenient to consider the normalizedintrinsic curvature R N = C V R which is obtained as R N = RC V = ( ∂ V P ) − T ( ∂ T,V P ) + 2 T ( ∂ V P )( ∂ T,T,V P )2( ∂ V P ) (2.4)According to the equation of state for VdW-like black holes (and VdW fluid), pressure P depends linearly on temperature T . It follows that ∂ T,T,V P = 0 and therefore the normalizedintrinsic curvature (2.4) reduces to R N = 12 (cid:104) − (cid:16) T ∂
V,T P∂ V P (cid:17) (cid:105) , (2.5)which is used to explore the microstructures of VdW-like black holes [34, 35].On the other hand, in order to observe the VdW type criticality, one needs to keepmost of thermodynamic quantities to be fixed, then the critical point depends on a fewerthermodynamic parameters. Now, it is interesting to see what happens if we fix one of thethermodynamic variables in the thermometric manifold to form a hypersurface and investigatethe critical behavior of the extrinsic curvature of such a particular hypersurface. In this regard,in Ref. [45] it has been found that the extrinsic curvature of a hypersurface of constantextensive/intensive variable in thermodynamic geometry is divergent at the phase transitionpoints and it also contains useful information about stability of a thermodynamic system.Let us focus on T = const isotherms in the P − V diagram. Form geometrical point ofview, it is equivalent to consider T constant hypersufaces in the thermodynamic manifolddetermined by metric elements (2.3). The unit normal vector of such hypersurfaces is givenby n i = ( n T , n V ) = (1 , and n i = g ij n j = ( √ C V T , . Thus the extrinsic curvature can bedefined by K = ∇ i n i and finally we obtain, K N = K (cid:112) C V = 12 (cid:104) − T ∂
V,T P∂ V P (cid:105) (2.6)In analogous with normalized intrinsic curvature, we have defined the normalized extrinsiccurvature K N . In the next sections, we investigate the behavior of these geometrical quantitiesnear the critical point for the VdW fluid and d dimensional charged AdS black holes . The first and simplest well-known example of an interacting system is the VdW fluid, whichcan be used to describe the first-order phase transition between gas and liquid phases. Theequation of state is given by P = Tv − b − av (3.1)where v is the specific volume of the fluid and is related to total volume by v = V /N where N the total number of all the microscopic molecules. The critical point is also determined– 4 –y using the set of two conditions ( ∂ v P ) T = 0 and ( ∂ v,v P ) T = 0 . By taking advantage ofequation of state (3.1), the critical point is P c = a b , v c = 3 b, T c = 8 a b (3.2)By defining the reduced variables, ˆ P = PP c , ˆ v = vv c , and ˆ T = TT c , then Eq. (3.1) becomes ˆ P = 8 ˆ T v − − v (3.3)which is the equation of state in the reduced parameter space [47]. Considering the reducedform of the number density ˆ n = 1 / ˆ v between the liquid and gas phases, one can write Eq.(3.3) in terms of the reduced fluid number density as ˆ P = 8 ˆ T ˆ n − ˆ n − n . (3.4)In the reduced parameter space, the isothermal compressibility κ T is also defined as κ T ≡ − V (cid:16) ∂V∂P (cid:17) T = − P c ˆ v (cid:16) ∂ ˆ v∂ ˆ P (cid:17) ˆ T = 1 P c ˆ n (cid:16) ∂ ˆ n∂ ˆ P (cid:17) ˆ T (3.5)Utilizing the expression for the reduced pressure in Eq. (3.3), Eq. (3.5) gives κ T P c = (3 − ˆ n ) / n T − ˆ n (3 − ˆ n ) (3.6)In order to expand the thermodynamic quantities around the critical point, we introduce thefollowing new variables for simplicity t = ˆ T − , ω = ˆ v − , π = ˆ p − . (3.7)One can obtain the critical exponent β which describes the behaviour of the liquid-gas numberdensity difference (order parameter) along ˆ p − ˆ T coexistence curve [47], ∆ n = n g − n l ∝ ( − t ) β for t < . (3.8)In Fig. 1, we depict ln( n g − n l ) as a function of ln(1 − ˆ T ) . The fitted straight line for thenumerical data points on the lower left with − ˆ T < − is determined by ∆ n = b (1 − ˆ T ) β with b = Exp (1 . . and β = 0 . which are in agreement with the criticalexponent β = 1 / and amplitude b = 4 predicted in [47]. On the other hand, the criticalexponent γ and γ (cid:48) determine the behavior of the isothermal compressibility κ T along theisochore ˆ n = 1 line and ˆ P − ˆ v coexistence curve, receptively, as follows κ T P c ∝ (cid:26) t − γ for t > , ( − t ) − γ (cid:48) for t < . (3.9)– 5 – itting :0.49999 Ln - T + - - - - - - - - - - Ln ( - T ) Ln ( n l - n g ) Figure 1 . Diagram of ln( n g − n l ) versus ln(1 − ˆ T ) on crossing the ˆ P - ˆ T gas-liquid coexistence curveof VdW fluid. The fitted straight line for the data points (red dot) is given by ∆ n = b ( − t ) β with b = 3 . and β = 0 . . Fitting : - - T - - - - - Ln ( - T ) Ln ( κ T p c ) (a) Fitting : - - T - - - - - Ln ( - T ) Ln ( κ T p c ) (b) Figure 2 . The Ln-Ln plot of κ T P c versus the difference − ˆ T . The slope of the fitted blue straightline for the numerical data, described filled red circles, (a) is -1.01239 along the coexistence saturatedliquid curve (b) is -0.98745 along the coexistence saturated gas curve. Therefore, by setting ˆ n = 1 in Eq. (3.6) and expanding in lower order of t , one can find γ = 1 .The ln-ln plot of the isothermal compressibility κ T P c versus (1 − ˆ T ) for t < along ˆ P − ˆ v coexistence curve is shown in Fig 2. The data are seen to follow the predicted asymptoticcritical behavior κ T P c ≈ ( − t ) γ (cid:48) with exponent γ (cid:48) = 1 in both coexistence saturated liquidand gas curves . Let us now examine the critical behavior of the intrinsic curvature R N andextrinsic curvature K N near the critical point. To do this, we need to rewrite Eq. (3.5) as Along ˆ P − ˆ v coexistence curve, it is suitable to write the κ T P c as a function of ˆ P and ˆ v . For this purpose,we employ the alternative definition of the isothermal compressibility as follows, κ T P c = − v (cid:16) ∂ ˆ v∂ ˆ P (cid:17) ˆ T = − v { ˆ v, ˆ T } ˆ v, ˆ P { ˆ P , ˆ T } ˆ v, ˆ P = 1ˆ v (cid:16) ∂ ˆ T∂ ˆ P (cid:17) ˆ v (cid:16) ∂ ˆ T∂ ˆ v (cid:17) − P . (3.10)Appx. A of Ref. [45] is devoted to a brief introduction to the bracket notation. – 6 –ollows (cid:16) ∂ ˆ P∂ ˆ n (cid:17) ˆ T = ( κ T P c ˆ n ) − . (3.11)With the help of the lever rule for the reduced volume, the reduced density in the coexistenceline can be also described by, n = θ ˆ n l + (1 − θ )ˆ n g , (3.12)where θ is the fraction of the liquid phase, so that for θ = 1 , we have the pure liquid phase( ˆ n = ˆ n l ) and for θ = 0 the pure gas phase ( ˆ n = ˆ n g ) will be dominated. Using the lever rulefor the reduce number density (3.12), Eq. (3.11) can be written as (cid:16) ∂ ˆ P∂ ˆ n (cid:17) ˆ T = 1 κ T P c (cid:16) θ ˆ n l + (1 − θ )ˆ n g (cid:17) . (3.13)By moving toward the critical point along the coexistence curve for t < , we therefore find (cid:16) ∂ ˆ P∂ ˆ n (cid:17) ˆ T (cid:12)(cid:12)(cid:12)(cid:12) n l = 1 κ T P c ˆ n l ∝ ( − t ) γ (cid:48) − β , (3.14) (cid:16) ∂ ˆ P∂ ˆ n (cid:17) ˆ T (cid:12)(cid:12)(cid:12)(cid:12) n g = 1 κ T P c ˆ n g ∝ ( − t ) γ (cid:48) − β . (3.15)As one approaches to the critical point along the isochore ˆ v = 1 ( ˆ n = 1 ) with t > , one alsoobtains (cid:16) ∂ ˆ P∂ ˆ n (cid:17) ˆ T = 1 κ T P c ∝ t γ . (3.16)On the other side, the expression inside the parentheses in Eqs. (2.5) and (2.6) can be writtenas T ∂
V,T P∂ V P = ˆ T ∂ ˆ V , ˆ T ˆ P∂ ˆ V ˆ P = ˆ T ∂ ˆ n, ˆ T ˆ P∂ ˆ n ˆ P = (1 + t ) ∂∂t (cid:16) ln (cid:16) ∂ ˆ P∂ ˆ n (cid:17) t (cid:17) . (3.17)Thus, the critical behavior of the above identity is determined by (cid:16) ˆ T ∂ ˆ n, ˆ T ˆ P∂ ˆ n ˆ P (cid:17) | n l ,n g ∼ ( γ (cid:48) − β ) t for t < , (3.18) (cid:16) ˆ T ∂ ˆ n, ˆ T ˆ P∂ ˆ n ˆ P (cid:17) | ˆ n =1 ∼ γt for t > . (3.19)Substituting these values into Eq. (2.5), the critical behaviors of R N for t → ± are obtained R N t ≈ − ( γ (cid:48) − β ) for t < , (3.20) R N t ≈ − γ for t > . (3.21)– 7 –n VdW fluid case with critical exponent β = 1 / and γ (cid:48) = γ = 1 , we have R N t ≈ − for t < , (3.22) R N t ≈ − for t > , (3.23)which exactly agrees with Refs. [34, 35] in t < case, whereas as t > , it is consistent withRefs. [48, 49] (when we take α = 0 ). Note that our choice of various trajectories in ˆ P − ˆ v diagram on approaching to the critical point leads to create a discontinuity in the value ofthe intrinsic curvature (3.22). In addition, the critical behavior of the normalized extrinsiccurvature (2.6) is given by K N t ≈ − γ (cid:48) − β for t < (3.24) K N t ≈ − γ for t > (3.25)By choosing critical exponents β = 1 / and γ (cid:48) = γ = 1 for the VdW fluid, one thereforearrives at K N t ≈ − for t < , (3.26) K N t ≈ − for t > , (3.27)Now, we can also numerically check this critical phenomena associated with the extrinsiccurvature. Near the critical point, one can numerically fit the formula by assuming that theextrinsic curvature K has the following form K ∼ − (1 − ˜ T ) − c K or ln | K | = − c K ln(1 − ˜ T ) + d K . (3.28)Regarding the intercept of the straight ln-ln lines for coexistence saturated liquid and gascurves in Fig. 3, we find K N t = K (1 − ˜ T ) (cid:112) C v = − (cid:114) e − . . = − . ≈ − (3.29)It seems that the amplitudes of the criticality of thermodynamic curvatures can be universalfor any VdW-like case. Therefore, we investigate the behavior of normalized curvatures nearthe critical point in VdW-like black holes such as four and higher dimensional charged AdSblack holes. In this section, we aim to check the universality of the relations (3.20) and (3.24) for a fourdimensional charghed AdS black hole. The thermodynamics and phase structure of such black The critical exponent α is for the heat capacity at constant volume. – 8 – itting : - - T - - - - - Ln ( - T ) Ln ( | K | ) (a) Fitting : - - T - - - - - Ln ( - T ) Ln ( | K | ) (b) Figure 3 . The extrinsic curvature ln | K | vs. ln(1 − ˜ T ) . The numerical data are given by the redmarkers and the fitting formulas are determined by blue solid lines. (a) Along the coexistence saturatedliquid curve, the slope is -0.99869. (b) Along the coexistence saturated gas curve, the slope is -1.00147. holes has been investigated in Ref. [6]. The equation of state for a 4D charged AdS black hole[6] is P = Tv − πv + 2 Q πv , (4.1)where the specific volume v = 2 r h ( r h is the radius of the event horizon.). It is interestingthat a phase transition between small and large black hole phases is the similar to the liquid-gas phase transition of the VdW fluid, where the critical point is given by P c = 1 / πQ , T c = √ / πQ , and v c = 2 √ Q . In the reduced parameter space, the equation of state reads ˆ P = 8 ˆ T v − v + 13ˆ v . (4.2)Moreover, the small/large black hole coexistence curve has the analytic form [51] as follows. ˆ T = ˆ P (3 − (cid:112) ˆ P ) / , (4.3)where ˆ T = T /T c and ˆ P = P/P c are reduced temperature and pressure, respectively. Theresult has been obtained by constructing the equal area law on an isobaric curve in the T − S diagram. By making use of this relation, one can obtain the reduced thermodynamic volumesfor small and large black holes along the coexistence curve, ˆ V s = (cid:113) − (cid:112) ˜ P − (cid:113) − (cid:112) ˆ P (cid:112) P , (4.4) ˆ V l = (cid:113) − (cid:112) ˆ P + (cid:113) − (cid:112) ˆ P (cid:112) P , (4.5)– 9 –here ˆ V = V /V c with V = πr and V c = 8 √ πQ . We now determine the critical behaviorof the difference in density between the small and large black hole phases on the coexistenceline. Using Eqs. (4.4) and (4.5), the asymptotic critical behavior of ∆ˆ n is given by ∆ˆ n = ˆ n l − ˆ n s ≈ √− t, (4.6)where n l ≈ √− t and n s ≈ − √− t for the charged AdS black holes. Comparisonof this expression with the definition in Eq. (3.8) gives the critical exponent β = 1 / . Thisexponent is the epitome of the mean-field theories of the second order phase transitions. Letus now consider the critical behaviors of κ T in the limit t → ± . Differentiating the pressure(4.2) with respect to the total volume gives (cid:16) ∂ ˆ P∂ ˆ V (cid:17) ˆ T = −
49 ˆ V / + 43 ˆ V / − T V / . (4.7)By writing the above relation in terms of the expansion parameters, i.e. t = ˆ T − and ω (cid:48) = ˆ V − and Taylor expanding to lowest orders, we have (cid:16) ∂ ˆ P∂ ˆ V (cid:17) ˆ T ≈ − t − ω (cid:48) for t > , (4.8) (cid:16) ∂ ˆ P∂ ˆ V (cid:17) ˆ T ≈ t − ω (cid:48) for t < . (4.9)Setting ω (cid:48) = 0 and using Eqs. (3.5) and (4.8), we get immediately κ T P c = 98 1 t for t > . (4.10)According to the definition in Eq. (3.9), it follows that γ = 1 . Up to the lowest order expansionin t , Eqs. (4.4) and (4.5) give w (cid:48) s,l ≈ t. (4.11)Substituting this value into Eq. (4.9) and using Eq.(3.5), we finally arrive at κ T P c = 916 1 t for t < . (4.12)Thus we can read the critical exponent γ (cid:48) from Eq. (3.9), i.e., γ (cid:48) = 1 . For the charged blackhole cases, Eqs. (2.5) and (2.6) give the normalized intrinsic curvature R N and extrinsiccurvature K N as R N = (3 ˆ V / − V / − − T ˆ V )2(1 − V / + 2 ˆ T ˆ V ) , (4.13) K N = 1 − V / − V / + 2 ˆ T ˆ V ) . (4.14)– 10 –riting these expressions in terms of the expansion parameters like t and ω (cid:48) where ω (cid:48) = 0 for t > and ω (cid:48) = ω (cid:48) s = − ω (cid:48) l = 3 √− t for t < , one can expand them up to lowest order in t asfollows. R N t ≈ (cid:26) − for t > , − for t < , and K N t ≈ (cid:26) − for t > , − for t < . (4.15)It is interesting that above results are in consistent with those for VdW fluid. More impor-tantly, the relations (3.20) and (3.24) tend to the same universal crirical amplitudes when onechooses γ = γ (cid:48) = 1 and β = 1 . For t < , we see that R N t ∼ − , which is the same asreported in Ref. [34]. Following the study in previous sections for the VdW fluid and 4D charged AdS black holes,we here attempt to investigate the critical behavior of R N and K N for a higher dimensionalcharged AdS black hole and to study the effect of the dimension of the spacetime on thisbehavior. In the reduced parameter space, the corresponding equation of state for higherdimensional RN-AdS black holes [35] becomes ˆ P = ˆ V − d − d − + ( d −
2) ˆ V − d (cid:16) − d + 4( d −
3) ˆ T ˆ V d − (cid:17) ( d − d − (5.1)Note that there is no analytic form of the coexistence curve for small-large black holes for thiscase in contrast with the four-dimensional case. However, highly accurate fitting formulas ofthe coexistence curves for d =5-10 have been obtained in Ref. [50]. Therefore, we numericallycalculate the coexistence curves very near the critical point, and then obtain the criticalbehaviour of our interesting quantities like such as isothermal compressibility, number density,normalized intrinsic curvature, and extrinsic curvature along the coexistence curves. In orderto calculate the critical exponents β , γ , and γ (cid:48) , let us first to study the treatment of theisothermal compressibility κ T P c and the number density difference ∆ n between the smalland large black holes near the critical point . The isothermal compressibility of the higherdimensional charged AdS black hole is calculated by using Eqs. (3.5) and (5.1) as follows. κ T P c = ( − d )( − d )( − d )2( d − (cid:18) ˆ V − d − (cid:16) d −
3) ˆ T ˆ V d − − d + 5 (cid:17) + ˆ V − d − d − (cid:19) . (5.2)On approach to the critical point along the isochore ˆ V = 1 with t > , the critical behaviourof the isothermal compressibility κ T is given by κ T p c ≈ ( d − d − d − t , (5.3) Note that critical exponents β and γ were calculated analytically in Ref. [52]. – 11 –hich implies that the critical exponent γ = 1 . Employing Maxwell’s construction, we numer-ically calculate the behaviour of ∆ n and κ T P c with t < for d =5-10 along ˆ P − ˆ T coexistencecurve and ˆ P − ˆ V coexistence curve, respectively, in Table 1.Quantity Coefficient d =5 d =6 d =7 d =8 d =9 d =10 c n ln(∆ n ) d n κ ln | κ T P c | (LBH) - d κ c κ ln | κ T P c | (SBH) - d κ Table 1 . Fitting values of the slope and intercept of ln ∆ n = c n ln(1 − ˆ T ) + d n and ln | κ T P c | = − c κ ln(1 − ˆ T ) − d κ straight line for coexistence saturated small-large black hole curves. According to the data appeared in Table. 1, it is obvious that the values of the criticalexponent β = c n and γ (cid:48) , which is the average of two slopes c κ (SBH) and c κ (LBH) , are notdeeply affected by increasing dimension of the spacetime, i.e., β ≈ , γ (cid:48) ≈ (5.4)which are the same as those for VdW fluid and 4D charged AdS black holes. Therefore, onecan expect that, independent of changing number of the dimension, the normalized intrinsiccurvature R N and extrinsic curvature K N show the universal critical behavior around thecritical point. Utilizing Eqs. (2.5), (2.6), and (5.1), one can easily find the criticality ofthermodynamic curvatures for t > are given by R N t ≈ − , K N t ≈ − , (5.5)which indicate a universal behaviour of thermodynamics curvatures near the critical pointsimilar to VdW fluid and 4D charged AdS black holes, i.e., they do not depend on the detailsof the physical system. Let us now examine the universality by approaching significantlycloser to the critical point along small/large black holes coexistence line. The coefficients ofthe numerical fitting of the ln-ln formulas of the R N and K N versus (1 − ˆ T ) have been collectedin Table 2.Taking numerical error into account, the slopes indicate that the critical exponent for R N and K N must be c R = 2 and c K = 1 , which is consistent with those of VdW fluid. Consideringthe interception of the straight ln-ln line (i.e., d R and d K ), the amplitude associated with thesecritical exponents are given in Table 3. – 12 –uantity Coefficient d =5 d =6 d =7 d =8 d =9 d =10 c R ln | R N | (SBH) - d R c R ln | R N | (LBH) - d R c K ln | K N | (SBH) - d K c K ln | K N | (LBH) - d K Table 2 . The coefficients of the numerical fitting of the ln-ln formulas for coexistence saturated smallblack holes (SBH) and coexistence saturated large black holes (LBH).
Quantity d =5 d =6 d =7 d =8 d =9 d =10 R N t -0.12506 -0.12507 -0.12508 -0.12509 0.12510 0.12511 − ( γ (cid:48) − β ) -0.12493 -0.12493 -0.12493 -0.12493 -0.12492 -0.12492 K N t -0.24982 -0.24982 -0.24984 -0.24985 -0.24986 -0.24987 − ( γ (cid:48) − β )2 -0.24993 -0.24993 -0.24993 -0.24993 -0.24992 -0.24992 Table 3 . Critical amplitudes of thermodyanic curvatures near the critical point.
Clearly, the change of amplitudes is negligible by increasing the number of spacetimedimensions. Therefore, we can conclude that the critical behavior of the normalized thermo-dynamic curvatures for t < along the coexistence curve are R N t ≈ − , K N t ≈ − . (5.6)which show universal behavior independently of the number of spacetime dimensions. More-over, they confirm universal amplitudes obtained in Eqs. (3.20) and (3.24). Note that in[34, 35], it is shown that the values of R N t depends on the spacetime number d , which isdifferent from that observed here. The reason is that the calculation accuracy in [34, 35] islower, and in this paper we only keep the lowest order of the expansion.– 13 – Conclusions and discussions
In this paper, we studied the critical behaviors of the normalized intrinsic curvature R N and the extrinsic curvature K N . The universal properties are analytically checked when thethermodynamic quantities are expanded to the lowest order.First, we dealt with the VdW fluid. Employing the equation of state, we found that R N has critical exponent 2 and K N has critical exponent 1 when the temperature parameter t approaches zero from two sides. The result of R N is also consistent with that of [34, 35]. Bymaking use of this result, we can construct the universal constants R N t and K N t . Adoptingthe numerical calculation, we find that R N t = − for t → − and − for t → + , whereas K N t = − for t → − and − for t → + .Next, we applied the treatment to charged AdS black holes. For the four dimensionalcharged AdS black hole, R N and K N , respectively, share the same critical exponents. Universalamplitude R N t = − for t → + and − for t → − and also, K N t = − and − for t → + and t → − , respectively.We also generalized the result to higher spacetime dimensions. When the temperaturetends to the critical value, the change of the number density ∆ n before and after the small andlarge black holes has critical exponent , which indicates it can act as an order parameter todescribe the small and large black hole phase transition. κ T P c also presents a critical exponent1 near the critical point when the black hole system approach the critical point along thecoexistence small and large black hole curves. Moreover, when t → + , we found R N t = − and K N t = − . And when t → − , R N t = − and K N t = − . Significantly, thesecritical amplitudes are independent of the spacetime number d . The numerical calculationsalso confirm this results.In summary, in this paper we have investigated the critical behaviors of the normalizedintrinsic and extrinsic curvature of the NTG geometry. Universal amplitudes at the criticalpoint are also calculated. These will uncover the underlying universal critical properties ofthe charged AdS black holes. The generalization to other AdS black hole backgrounds arealso expected. Acknowledgements
We are grateful to Yu-Xiao Liu and B. Mann for their careful reading and extremely helpfuldiscussions and comments on this work. Shao-Wen Wei was supported by the National NaturalScience Foundation of China (Grant No. 11675064) and the Fundamental Research Funds forthe Central Universities (Grants No. lzujbky-2019-it21).
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