The thermodynamic relationship between the RN-AdS black holes and the RN black hole in canonical ensemble
aa r X i v : . [ h e p - t h ] A p r EPJ manuscript No. (will be inserted by the editor)
The thermodynamic relationship between the RN-AdS blackholes and the RN black hole in canonical ensemble
Yu-Bo Ma , Li-Chun Zhang , Jian Liu , Ren Zhao , Shuo Cao Department of Astronomy, Beijing Normal University, Beijing 100875, China; Institute of Theoretical Physics, Shanxi Datong University, Datong 037009, China; School of Physic, Shanxi Datong University, Datong, 037009, China; School of Physic, The University of Western Australia,Crawley, WA 6009, Australiathe date of receipt and acceptance should be inserted later
Abstract.
In this paper, by analyzing the thermodynamic properties of charged AdS black hole and asymp-totically flat space-time charged black hole in the vicinity of the critical point, we establish the correspon-dence between the thermodynamic parameters of asymptotically flat space-time and nonasymptoticallyflat space-time, based on the equality of black hole horizon area in the two different space-time. The re-lationship between the cavity radius (which is introduced in the study of asymptotically flat space-timecharged black holes) and the cosmological constant (which is introduced in the study of nonasymptoticallyflat space-time) is determined. The establishment of the correspondence between the thermodynamics pa-rameters in two different space-time is beneficial to the mutual promotion of different time-space blackhole research, which is helpful to understand the thermodynamics and quantumproperties of black hole inspace-time.
PACS.
XX.XX.XX No PACS code given
The AdS black hole solution in four-dimensional space-time is an accurate black hole solution of the Einsteinequation with negative cosmological constant in asymp-totic AdS space time [1]. This solution has the same ther-modynamic characteristics as the black hole solution inasymptotically flat space-time, i.e., the black hole entropyis equal to a quarter of the event horizon area, while thecorresponding thermodynamics quantity satisfies the lawof thermodynamics of black hole. It is well known that,if taken as a thermodynamic system, the asymptoticallyflat black hole does not meet the requirements of thermo-dynamic stability due to its negative heat capacity. How-ever, compared with the asymptotically flat space-timeblack hole, the AdS black hole can be in thermodynamicequilibrium and stable state, because the heat capacity ofthe system is positive when the system parameters takecertain values.Therefore, the thermodynamics of AdS charged blackholes, in particular its phase transition in (n+1)-dimensionalanti-de Sitter space-time was firstly discussed and exten-sively investigated in Refs. [2, 3], which discovered thefirst order phase transition in the charged non-rotatingRN-AdS black hole space-time. Recently, increasing atten-tion has been paid to the possibility that the cosmological a e-mail: [email protected] constant Λ could be an independent thermodynamic pa-rameter (pressure), and the first law of thermodynamics ofAdS black hole may also be established with P − V terms.For instance, the P − V critical properties of AdS blackhole was firstly studied in Ref [4], which found that thephase transition and critical behavior of RN-AdS blackhole are similar to those of the general van der waals-Maxwell system. More specifically, the RN-AdS black holeexhibits the same P − V (liquid-gas phase transition) crit-ical phase transition behavior and critical exponent as vander waals-Maxwell system. The phase transition and crit-ical behavior of various black holes in the extended phasespace of AdS have also been extensively studied in the lit-erature [5–23], which showed very similar phase diagramsin different black hole systems.The asymptotically flat black holes cannot reach ther-modynamic stability, due to the inevitable so-called Hawk-ing radiation. In order to obtain a better understandingof the thermodynamic properties and phase transition ofblack holes, we must ensure that the black hole can achievestability in the sense of thermodynamics. According tothe previous results obtained by York et al.(1986) [24],achieving thermodynamic stability for asymptotically flatblack hole system also depends on the effect of environ-ments, i.e., one needs to consider the ensemble system. Dif-ferent from the general thermodynamic system, the self-gravitational system has inhomogeneity in space, which Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle makes it necessary to determine the corresponding ther-modynamic quantities and their characteristic values.The local thermodynamic stability of self-gravitationalsystems can be analyzed by considering the extreme valueof the Helmholz free energy of the system. When it reachesto a minimum value, the corresponding system is at leastlocally stable. According to the methods extensively stud-ied in the literature [25–31], the extreme value of the freeenergy of gravitational systems can be derived from theaction I, i.e., the partition function of the system at thezero-order approximation can be calculated by using theGibbons-Hawking Euclidean action [32] Z ≈ e − I E (1)Combining this with the Helmbolz free energy F from theequation Z = e − βF , we can obtain I E ( r, T, Q ; r + ) = βF = βE ( r, Q ; r + ) − S ( r + ) (2)where r , T , Q respectively denote the radius, tempera-ture and electric charge of the cavity, β = 1 /T , and r + is the radius of the black hole horizon. E ( r, Q ; r + ) and S ( r + ) are the internal energy and entropy of the blackhole in the cavity. Therefore, when the r, T, Q is deter-mined, the only variable for the system is r + . The thermal-equilibrium conditions of the black hole and environmentcan be determined by the following equation dI E dr + (cid:12)(cid:12)(cid:12)(cid:12) r + =¯ r + = 0 (3)The conditions that the free energy reaches to its mini-mum value is that the system is at local equilibrium state.In order to reach to the thermal equilibrium, it should sat-isfy the following criteria d I E dr (cid:12)(cid:12)(cid:12)(cid:12) r + =¯ r + > . (4)Utilizing the method described above, the literature [26–32] have studied the charged black hole and black branes,obtained the requirements to meet the thermodynamicequilibrium conditions, and discussed the phase transi-tion and critical phenomena. More recently, Eune et al.(2015) investigated the phase transition based on the cor-rections of Schwarzschild black hole radiation temperature[33]. More specifically, both of the charged black hole andthe radiation field outside the black hole were consideredin their work, under the condition that they are both inthe equilibrium state.On the other hand, Reissner-Nordstrom (RN) blackhole and RN-AdS black hole are the exact solutions ofthe Einstein equation. The main difference between AdSspace-time and asymptotically flat space-time is the fa-mous Hawking-Page phase transition [1], i.e., the AdSbackground provides a natural constraint box, which makesit possible to form a thermal equilibrium between largestable black hole and hot gas. In the recent study of thephase transition of the RN black hole, in order to meet the requirement of thermodynamic stability, one needs to arti-ficially add a cavity concentric in the horizon of the blackhole. However, the determination of the specific value ofthe radius of the cavity is still to be done. In the previ-ous studies of phase transient of RN and RN-AdS blackholes, it was found that both of the two types of blackholes exhibit the same P − V (liquid-gas phase transi-tion) properties as van der Waals-Maxwell system, whichis also consistent with our finding through the compari-son between the phase transition curves of these two kindsof black holes. Therefore, the consistency of the thermo-dynamic stable phase and phase transition between thecavity asymptotically flat black hole and the black hole inthe AdS space seems to indicate a more profound conno-tation [25, 26]. One of the roles played by the cavity andthe AdS space is to ensure the conservation of the degreeof freedom within a certain system. Naturally, the discus-sion of the following problems is the main motivation ofour analysis: Does a duality of a gravitational theory anda non-gravitational theory exist in the cavity setting? Arethese two types of black holes inherently connected? If so,can the cavity radius introduced for the RN black holebe determined by the thermodynamic properties of theRN-AdS black hole?In this paper, by comparing the phase transition curvesof the RN black hole with the RN-AdS black hole, we es-tablish the equivalent thermodynamic relations of the twokinds of black holes. We also discuss the relationship be-tween the radius of cavity introduced into the RN blackhole, the black hole horizon radius, and the cosmologicalconstant. Finally, we investigate the equivalent thermody-namic quantities of two kinds of black holes, which pro-vides theoretical basis for the further exploration of theirinternal relation. To begin with, we review the thermodynamic proper-ties of RN black holes. The metric of a charged RN blackhole is given by ds = − V ( r ) dt + dr V ( r ) + r dΩ (5)where V ( r ) = 1 − Mr + ¯ Q r . (6)The corresponding action expresses as [21, 25] I E ( r B , T B , ¯ Q ; r + ) = β B F = β B E ( r B , ¯ Q ; r + ) − S ( r + )= β B r B − s(cid:18) − r + r B (cid:19) (cid:18) − ¯ Q r B r + (cid:19)! − πr , (7)Here T B is the temperature of the cavity and β B = 1 /T B . r B is the radius of concentricity outside a black hole hori- lease give a shorter version with: \authorrunning and \titlerunning prior to \maketitle zon r + , which can be obtained from the following equation V ( r + ) = 1 − Mr + + ¯ Q r = 0 (8)The corresponding reduction quantities are defined as¯ I E = I E πr B , x = r + r B , q = ¯ Qr B , ¯ b = β B πr B (9)Note that the relation of q < x < r + > ¯ Q , r B > r + ) and the reduced action takes the formas ¯ I E (¯ b, q, x ) = ¯ b − s (1 − x ) (cid:18) − q x (cid:19)! − x . (10)Therefore, we can obtain d ¯ I x dx = 1 − q x − x ) / (cid:16) − q x (cid:17) / (cid:0) ¯ b − b q ( x ) (cid:1) , (11)where the function of the reciprocal of temperature is b q ( x ) = x (1 − x ) / (cid:16) − q x (cid:17) / − q x . (12)The condition for the RN Black hole and cavity to reachthermal equilibrium is d ¯ I E dx = 0 ⇒ ¯ b = b q (¯ x ) . (13)Thus the reciprocal of the black hole radiation tempera-ture can be written as1 T = 4 πr B b q ( x ) = 4 πr B x (1 − x ) / (cid:16) − q x (cid:17) / − q x . (14)The critical charge and critical radius of a black holecan be determined by the following conditions db q ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) x = x c = 0 d b q ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) x = x c = 0 (15)Combing Eqs. (12) and (15), one can easily derive thecritical charge and the critical radius of black hole as [25] q c = √ − x c = 5 − √ b q ( x ) with respect to x is shown in Fig. 1. P − V criticality of charged AdS blackholes To start with, we review some basic thermodynamicproperties of the spherical RN-AdS black holes. In the framework of Schwarzschild-like coordinates, the metricand the U (1) field read ds = − f ( r ) dt + dr f ( r ) + r dΩ (17) F = dA, A = − Qr dt (18)where dΩ stands for the standard element on S and thefunction f is given by f = 1 − Mr + Q r + r l . (19)where l is the AdS length scale and is related to the cos-mological constant as Λ = − /l . The radius of the blackhole horizon, r + , satisfies the following equation f ( r + ) = 1 − Mr + + Q r + r l = 0 (20)and the black hole radiation temperature reads T = 14 πr + (cid:18) r l − Q r (cid:19) . (21)In order to explicitly illustrate the critical phenomenon ofRN black hole, we present the β − r + diagram in Fig. 2,from which the phase-transition point of system could beobtained as Q c = l , r c = l √ , β c = πl √ . (22)These results are well consistent with those obtained inthe previous analysis [4].Moreover, the state parameters of a certain systemshould satisfy the first law of thermodynamics dM = T dS + ΦdQ + V dP (23)where the potential Φ , thermodynamic volume V and pres-sure P of the black hole respectively express as Φ = Qr + , V = 43 πr , P = 38 πl . (24) As can be seen from Fig. 1 and 2, for the two typesof black holes, the curves of the reciprocal of tempera-ture with respect to the radius of black hole horizon arequite similar to that for the Van der Waals-Maxwell gas-liquid phase transition (the corresponding critical expo-nents can be calculated at the critical point). Moreover,similar to the cases in the AdS space and dS space, thespecific heat capacity of the black hole in the asymptoti-cally flat space can be expressed as c v ∼ ( T − T c ) − / [29], Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle x b q H x L Fig. 1.
The characteristic behavior of b q ( x ) as a function of x , when q = q c − ∆q ( red ) , q = q c ( blue ) , q = q c + ∆q ( black ), with q c = √ − and ∆q = 0 . . The blue curve in the middle represent the critical curve. r + Β Fig. 2. β − r + diagram when Q = Q c - ∆Q ( black ), Q = Q c ( blue ), Q = Q c + ∆Q ( red ), with Q c = l , l = √ , and ∆Q = 0 . . The blue curve in the middle represent the critical curve. where T c denotes the critical temperature. These commonthermodynamic properties hint us the possibility to makethe thermodynamic properties of the black hole in thesespace-time well consistent with each other, only by ad-justing some specific parameters. Therefore, a cavity ra-dius, which may ensure the thermodynamic equilibriumstability of a system, is introduced in our discussion ofthe thermodynamic properties of charged black holes inthe flat space-time. Meanwhile, there is no definite valuefor the radius in our analysis, which provides us the pos-sibility to adjust this parameter to obtain similar thermo-dynamic properties from the black holes in two differentspace-time, and finally derive the relationship between theradius of the cavity and the radius of the black hole r + orthe cosmological constant l .It is generally believed that, if the corresponding stateparameters in two thermodynamic system behave the sameas the selected independent variables, these two systemswill have the same thermodynamic properties. In orderto obtain similar thermodynamic properties of the blackholes in two different space-time, we ensure that with thechange of horizon radius of black hole, the temperatureand entropy of the two systems are kept to be equal. Thatis, the radii of the two black holes are equal to each other in two different space-time, which leads to the similar en-tropy in the two systems. In the vicinity of the criticalpoint, the temperature of a black hole in the space-timevaries synchronously with the radius of the horizon. Basedon the assumption that the entropy and temperature ofa black hole in two different space-time change the samewith the horizon radius, we can obtain the relationshipbetween the radius of the cavity introduced in the flatspace-time and the radius of the black hole horizon andthe cosmological constant.Supposing the radii of the black hole horizon in the twodifferent space-time are both r + , which can be respectivelydetermined by Eq. (8) and Eq. (20) for RN black holesand RN-AdS black holes. The common entropy of thesetwo systems expresses as S = πr (25)As can be seen from the analysis in Section. 3, both ofthe two types of black holes show the same characteristicsas Van der Waals-Maxwell system (fluid-gas phase transi-tion), the critical points of which are respectively given byEqs. (16) and (22). It is apparent that, Eq. (25) implies thecritical point of RN black holes and RN-AdS black holesare determined when the black hole radius is the same in lease give a shorter version with: \authorrunning and \titlerunning prior to \maketitle two different space-time, while Eq. (22) shows the criti-cal value of the RN-AdS black hole is dependent on AdSlength scale l . Therefore, when the horizon position r + andcharge Q are given for such black hole, the radiation tem-perature depends on the cosmological constant. On theother hand, it can be seen from Eq. (13) that, when q and x are fixed, the temperature of RN black holes dependson the cavity radius r B .To start with, when the black hole horizon r + is thesame in the two different space-time, we should determinethe requirement of coincidence of critical curves, which isalso the requirement to make T = T . With the definitionof y = r + /l , the combination of Eq. (13) and (21) gives1 − y + 3 y = 1 − q c /x √ − x q − q c x (26)to obtain equal temperature in two time-space, where Q c = l , q c = √ −
2. The quantitative relation between x and y is plotted in Fig. 3. Because the black hole horizon po-sition r + is the same for both types of black holes, the x − y curve also quantifies the relation between l and r B .We can get y = y c = 0 . x = x c = 5 − √ β ( x ) − x of RN black holeand β ( y ) − x of RN- AdS black hole coincide the criticalcurve of Black Hole.Secondly, when T = T , at the critical point y = y c , x = x c , supposing there is a deviation of the chargeto the threshold, we can get ∆Q = l∆ ˜ Q and ∆q from thefollowing expression (cid:18) r l − Q r (cid:19) = 1 − q /x √ − x q − q x (27)It is straightforward to obtain − Q c ∆Qr = − ∆ ˜ Q y c = 1 √ − x c (cid:16) − q x c (cid:17) / q c x c (cid:18) − x c + q c x c (cid:19) ∆q (28)Substituting y c = 0 . x c = 5 − √ ∆ ˜ Q and ∆q and the Eq. (27) can be rewritten as y − y − ∆ ˜ Q y ! = 1 − q /x √ − x q − q x (29)When ∆ ˜ Q is given, one can get the value of ∆q fromEq. (28). Denoting q = q c + ∆q = √ − ∆q in Eq. (29),we can obtain the y − x curve deviating from the criticalcharge. The results are plotted in Fig. 4, in which the blue,red and black lines respectively correspond to the threecases with the critical charge, above the critical charge andunder critical charge. It is obvious that when the relationbetween y and x satisfies Eqs. (26) and (29), the critical curves β ( x ) − x of RN black holes and β ( y ) − y of RN-AdSblack holes coincide, which implies the equal temperatureof the two different space-time. Therefore, similar to thedefinition of entropy in Eq. (25), the heat capacity of thetwo systems can also be written as C Q = T (cid:18) ∂S ∂T (cid:19) Q = C ¯ Q = T (cid:18) ∂S ∂T (cid:19) ¯ Q (30)We remark here that in the discussion above, the cavityradius is introduced as a state parameter to test the ther-modynamic properties near the critical point of the RNblack hole in the asymptotically flat space-time. This pro-cedure follows the study of the thermodynamic propertiesnear the critical point of the RN-AdS black hole, wherethe cosmological constant is taken as the state parameterin the thermodynamic system. Our results demonstratesthat the change of the two systems is the same near thecritical point, as long as the variables y and x introducedin the two system satisfy Eqs. (26) and (29). As is shownin Figs. 1-2, the radius of the cavity is a one-to-one corre-spondence with the cosmological constant, and thus couldbe the dual of the cosmological constant in non-asymptoticflat space-time. It is well known that black hole is an ideal systemto understand the nature and behavior of quantum grav-ity. On one hand, black hole provides an ideal model tostudy all kinds of interesting behaviors of classical gravi-tation (under the sense of general relativity); on the otherhand, it can be regarded as a macroscopic quantum sys-tem with unique thermodynamic properties (the entropy,temperature and holographic properties of gravitation arequantum), which provides an important window to probethe quantum gravity. More importantly, a better under-standing of the black hole singularity, cosmological singu-larity, and cosmological inflation needs a basic theory ofspace-time gravitation. Up to now there is still no maturetheory to precisely describe the quantum characteristicsof the asymptotic flat black holes and non-asymptoticallyflat space-time black holes. However, string theory firstlyprovided the microcosmic explanations for the entropy ofsome AdS black holes, which predicted the existence of M-theory and some duality relations (especially AdS / CFTcorrespondence) and realized the holographic propertiesof gravitational systems [35].In this paper, by setting the black-hole horizon at thesame value and assuming the temperature near the criti-cal point is equal in two space-time, we have discussed therelationship between the radius of the cavity r B and thecosmological constant l . The former parameter is alwaysintroduced in studying the thermodynamics of the chargedblack hole in the flat space-time, while the latter is relatedto the non-flat space-time. Our results may provide a the-oretical basis for exploring the internal relations of ther-modynamics in the black space. Moreover, by establishing Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle x y Fig. 3.
The y − x curve for Eq. (26) with Q c = l , q c = √ − , l = √ x y Fig. 4.
The y − x curves for ∆ ˜ Q = 0 .
05 (blue), ∆ ˜ Q = 0 (red), and ∆ ˜ Q = − . black ). the correspondence between the thermodynamic param-eters of black holes in the asymptotically flat space-timeand non-asymptotically flat space-time, we hope to seekthe correspondence between the quantum properties of theblack hole in two different space-time, which is beneficialto understand the thermodynamics and quantum proper-ties of black holes in different space-time. Acknowledgments
The authors declare that there is no conflict of inter-est regarding the publication of this paper. This work wassupported by the Young Scientists Fund of the NationalNatural Science Foundation of China (Grant Nos.11605107and 11503001), in part by the National Natural ScienceFoundation of China (Grant No.11475108), Supported byProgram for the Innovative Talents of Higher LearningInstitutions of Shanxi, the Natural Science Foundationof Shanxi Province,China(Grant No.201601D102004) andthe Natural Science Foundation for Young Scientists ofShanxi Province,China (Grant No.201601D021022), theNatural Science Foundation of Datong city(Grant No.20150110).
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