Phase Transition and Clapeyon Equation of Black Hole in Higher Dimensional AdS Spacetime
aa r X i v : . [ h e p - t h ] J un Phase Transition and Clapeyron Equation of Black Holes inHigher Dimensional AdS Spacetime
Hui-Hua Zhao, Li-Chun Zhang, Meng-Sen Ma, Ren Zhao ∗ Institute of Theoretical Physics, Shanxi Datong University, Datong 037009, China andDepartment of Physics, Shanxi Datong University, Datong 037009, China
By Maxwell equal area law we study the phase transition in higher dimensionalAnti-de Sitter (AdS) Reissner-Nordstr¨om (RN) black holes and Kerr black holes inthis paper. The coexisting region of the two phases involved in the phase transitionis found and some coexisting curves are shown in P − T figures. We also analyticallyinvestigate the parameters which affect the phase transition. To better compare witha general thermodynamic system, the Clapeyron equation is derived for the higherdimensional AdS black holes. Keywords : Reissner-Nordstr¨om AdS black hole, Kerr AdS black hole, phasetransition, two phase coexistence, Clapeyron equation
PACS numbers: 04.70.-s, 05.70.Ce
I. INTRODUCTION
Black hole is an ideal platform for studying many interesting behaviors in classical gravitytheory. It also can be regarded as a macroscopic quantum system in that its peculiarthermodynamic properties and thermodynamic quantities, as entropy, temperature, and itsgravity holographic nature are quantum mechanical intrinsically. Hence research of blackhole thermodynamics is an important channel for studying quantum gravity [1–5]. Althoughthermodynamics of black hole has been researched for many years, the exact statisticalinterpretation for thermodynamics state of black hole is not clear. So the thermodynamicsof black hole remains an important subject.It has been found that a black hole possesses not only standard thermodynamic quantities ∗ corresponding author: Email:[email protected](Ren Zhao) but abundant phase structures and critical phenomena, such as those involved in Hawking-Page phase transition[6], similar to the ones of a general thermodynamic system. Evenmore interesting is that the studies on the charged black holes show they may have ananalogous phase transition with that of van der Waals-Maxwell liquid-gas[7, 8]. Recentlycritical behaviors and phase transition of black holes have been extensively investigatedby considering the cosmological constant as thermodynamic pressure[9–13], P = − π Λ = ( d − d − πl . People have been trying to construct a complete liquid-gas analogue system forblack holes.In ref.[9, 10, 14–19], the phase transition and critical behaviors of some AdS black holeshave been studied, which exhibits several different phase transition behaviors. On the basisof the thermodynamic volume of dS spacetime given in ref.[11, 13, 20], considering theconnection between black hole horizon and cosmological horizon, we derived the effectivetemperature and effective pressure in some dS spacetimes in our previous works[21–24]. Therelation among the effective thermodynamic quantities was investigated, which shows thecritical phenomena like that in van der Waals liquid-gas system. Using Ehrenfest scheme, thesecond order phase transition was proved to exist in AdS spacetime black hole at the criticalstates[14, 17, 25–32]. Alike conclusion has been reached by investigating thermodynamicsand state space geometry of black holes in ref.[14, 32–37]. The phase transition behaviorsof black holes is found to be related to not only the spacetime metric but also the theory ofgravity or other factors[31, 38–42].Although some encouraging results have been achieved about the thermodynamics ofblack holes in AdS and dS spacetimes, many thermodynamic properties need to be inves-tigated more specifically. It is significant to study the critical behaviors and the process ofnoncritical phase transition in detail. The analyses on the P ∼ v relation of some black holesin AdS spacetimes show a negative pressure appear as the temperature is below a certainvalue and a thermodynamic unstable region exists with ∂P/∂v > P − v plots. We expect to provide some particularinformation about phase transition and properties of black hole thermodynamic system inAdS spacetimes to contribute to search for more stable black holes and to explore propertiesof quantum gravity.The paper is arranged as follow: the thermodynamic quantities of the d -dimensional RNAdS black hole and the d -dimensional Kerr AdS black hole are introduced firstly in section2 and in section 3 respectively. Then the phase transition of the black holes are studied byMaxwell equal area law, and the effect of the dimension and the thermodynamic quantities,as electric charge in section 2 and angular momentum in section 3, on the phase transition isanalyzed . In section 4, we make some discussion. (we use the units G d = ~ = k B = c = 1) II. THE CHARGED BLACK HOLESA. Thermodynamics
Reissner-Nordstrom black holes are characterized by the spherically symmetry and electri-cal nature. The solution for d -dimensional RN-AdS spacetime with a negative cosmologicalconstant, Λ = − ( d − d − / l , is defined by the line element[7] ds = − f dt + f − dr + r d Ω d − , (2.1)where f = 1 − mr d − + q r d − + r l . (2.2)The ADM mass and the electric charge have been defined as ( G d = 1), M = ω d − ( d − π m, Q = ω d − p d − d − π q, (2.3)in which the volume of the unit d -sphere ω d can be expressed as ω d = 2 π ( d +1) / Γ (cid:0) d +12 (cid:1) . (2.4)The corresponding Hawking temperature, entropy, and electric potential of the system aredefined as T = d − πr + − q r d − + d − d − r l ! , S = ω d − r d − , Φ = s d − d − qr d − . (2.5)In our consideration we interpret the cosmological constant Λ as a thermodynamic pressure P [13], P = − Λ8 π = ( d − d − πl . (2.6)The corresponding conjugate quantity, the thermodynamic volume, is given by[13] V = ω d − r d − d − r + being the position of black hole horizon determined from f ( r + ) = 0. CombiningEqs. (2.5) and (2.6), we can obtain P = T ( d − r + − ( d − d − πr + q ( d − d − πr d − . (2.8)Comparing (2.8) with the Van der Waals equation, we conclude that the specific volume v should be identified with the horizon radius of the black hole as[9, 10] v = 4 r + l d − p d − . (2.9)In geometric units we have r + = kv, k = d − , (2.10)and the equation of state reads P = Tv − ( d − π ( d − v + q ( d − πv d − k d − . (2.11)In Fig.1, the isotherms are plotted in P − v diagrams for different d and q . It shows thereare thermodynamic unstable regions with ∂P/∂v > T < e T . The isotherm corresponding to T = e T is tangent to horizontal axis with P = 0, in which ˜ T can be obtained by e T ( d − r + − ( d − d − πr + q ( d − d − πr d − = 0 , (cid:18) ∂P∂v (cid:19) ˜ T = 0 . (2.12) e T = 2( d − π ( d − d − v . (2.13) B. Maxwell equal area law and two phase equilibria
The state equation of the d -dimensional RN-AdS black hole is exhibited in the P − v diagrams in Fig.1, which show the existence of negative pressure and thermodynamic
10 20 30 40 50 v - - P (a) d = 4; q = 2, 3, 4 v - - P (b) d = 8; q = 1, 3, 5 v - - P (c) d = 10; q = 1, 3, 5 FIG. 1:
Isotherms in P − v diagrams. In each diagram the three groups of curves correspond tothe three given values of q , the lower ones (red curves) correspond to the smallest q , the middleones (blue curves) meet with the medium q , and the upper ones (green curves) are with the biggest q . The five curves in every group represent five different temperatures, the lower curve the lowertemperature. unstable region with ∂P/∂v >
0, which makes a thermodynamic system contract and expandautomatically in classical thermodynamics. The case occurs in van der Waals equation,where the problems have been resolved by reference to some practical process of phasetransition in real fluid and by Maxwell equal area law.Using Maxwell equal area law to deal with the state equation of the d -dimensional RN-AdS black hole, the two-phase equilibrium lines, where the two phases coexist, are acquired.Take T c as critical temperature of the d -dimensional RN-AdS black hole, and supposeat temperature T ( T ≤ T c ) the boundary state parameters for two-phase coexistence are( v , P ) and ( v , P ). According to Maxwell equal area law, P ( v − v ) = v Z v P dv, (2.14)we can get P ( v − v ) = T ln (cid:18) v v (cid:19) + ( d − π ( d − (cid:18) v − v (cid:19) − q ( d − d − πk d − (cid:18) v d − − v d − (cid:19) , (2.15) P = T v − ( d − π ( d − v + q ( d − πv d − k d − , P = T v − ( d − π ( d − v + q ( d − πv d − k d − . (2.16)Set x := v /v , from Eqs. (2.15) and (2.16), we can obtain v d − = q y ( x ) y ( x ) , (2.17)where y ( x ) = ( d − k d − (cid:18) (1 − x d − ) ln x + (cid:20) (1 − x d − )2 d − − x d − (cid:21) (1 − x ) (cid:19) ,y ( x ) = 2( d − d − x d − (1 − x ) ((1 + x ) ln x + 2(1 − x )) . (2.18)While x →
1, from (2.17) we get the critical specific volume v c = 1 k (cid:2) q ( d − d − (cid:3) / (2 d − . (2.19)According to (2.16), T v d − x d − = ( d − π ( d − v d − x d − (1 + x ) − q ( d − πk d − (1 − x d − )1 − x , (2.20)and the critical temperature and critical pressure are obtained as x → T c = ( d − πv c k (2 d − , (2.21) P c = ( d − πv c k . (2.22)These critical thermodynamic quantities are consistent with those in Ref.[10].Combining (2.17), (2.20) and (2.21), as T = χT c with χ ≤
1, we can get χ ( d − x d − (2 d − d − d − / [2( d − (cid:18) y ( x ) y ( x ) (cid:19) (2 d − / (2 d − = (1 + x )( d − x d − (cid:18) y ( x ) y ( x ) (cid:19) − k d − (1 − x d − )1 − x . (2.23)For a fixed χ , i.e. a fixed T , we can evaluate x with (2.23), and then according to (2.17)and (2.16), the v , v and P are specified while q is determined. Join the points ( v , P )and ( v , P ) on the isotherm with T = T in the P − v diagram, which generate an isobarrepresenting the process of phase transition like that of van der Waals system. Fig.2 showsthe isobars with solid (red) straight lines and the boundary of the region of two phasecoexistence by the dot-dashed (green) curve. In Fig.2 each combined solid line simulates theprocess of thermodynamic state change and phase transition of d -dimensional RN-AdS black v P d = q = FIG. 2:
The simulated phase transition(red solid lines) and the boundary of two phase coexistence(green dot-dashed curve) on the base of isotherms in P − v diagram for RN-AdS black hole with d = 8 , q = 5 . hole at a certain temperature. The phase transition process becomes shorter as temperaturegoes up until it turns into a single point at a certain temperature, which is the criticaltemperature, and the point corresponds to critical state of d -dimensional RN-AdS blackhole.When taking χ = 0 . , . , . , . , .
9, we have calculated the quantities x , v , P as q =0.5, 1, 1.5 in the spacetimes with the dimension d = 4 , ,
10 respectively. The resultsare shown in Table 1.From Table 1, it can be seen that x is unrelated to q , but incremental with the increase of χ / d at certain d / χ . v decreases with the increasing χ / d and increases with the incremental q when the other parameters are fixed respectively . P increases with the incremental χ / d and decreases with the increasing q with others determined parameters respectively. C. P − T curves of two phase coexistence From isothermal P − v curves in Fig.1, we know that when temperature T < ˜ T , thenegative pressure section on the isotherm emerges and becomes larger with lower tempera-ture. From Fig.2, we can see that the lower temperature, the smaller the value of P , which TABLE I:
The parameters of two phase coexistence for RN-AdS black hole. d = 4 d = 8 d = 10 q χ x v P x v P x v P . . also can be seen in Table 1. The doubt is whether P is negative when temperature is lowenough.Considering (2.17), (2.16) can be rewritten as P q / ( d − (cid:18) y ( x, ) y ( x ) (cid:19) ( d − / ( d − = χ ( d − π (2 d − d − d − / [2( d − (cid:18) y ( x ) y ( x ) (cid:19) (2 d − / (2 d − − ( d − π ( d − (cid:18) y ( x, ) y ( x ) (cid:19) + ( d − πk d − (2.24)From (2.23), (2.24) and T = χT c , one can get the relation between P and T , which isshown in Fig.3. Fig.3 exhibits the P − T phase diagrams at fixed q and d , in which thecurves represent the states of two phase coexistence and the terminal points are the criticalpoints. From Fig.3 it can be seen that the influence of the electric charge q and spacetimedimension d on the phase diagrams, however pressure P tends toward zero with decreasingtemperature T for all of the fixed q and d cases. That the pressure P is always positivemeans Maxwell equal area law is appropriate to resolve the doubts about negative pressureand unstable states in phase transition of the d -dimensional RN-AdS black hole. In Fig.4,the P q / ( d − − χ connection is depicted at different spacetime dimension d , which explicitlyexhibits the effect of the dimension d on the distribution of the black hole phases. T P q = q = q = d = (a) d = 4; q = 1, 3, 5 T P q = q = q = d = (b) d = 8; q = 1, 3, 5 T P q = q = q = d = (c) d = 10; q = 1, 3, 5 FIG. 3: P − T phase diagrams at fixed q and d for d -dimensional RN-AdS black hole. From (2.15)-(2.17), one can get P = y ( x ) , T = y ( x ) , (2.25) Χ P q - d = d = d = FIG. 4: P q / ( d − − χ phase diagram for d -dimensional RN-AdS black hole with d = 4 , , y ( x ) = d − π (cid:20) d − x − − x d − k d − x d − (1 − x ) y ( x ) y ( x ) (cid:21) (cid:18) y ( x ) y ( x ) q (cid:19) d − ,y ( x ) = ( d − π (cid:20) x ( d − x − − x d − k d − x d − (1 − x ) y ( x, ) y ( x ) (cid:21) (cid:18) y ( x, ) y ( x ) q (cid:19) d − . (2.26)From (2.25) dP dT = y ′ ( x ) y ′ ( x ) (2.27)with y ′ ( x ) = dy/dx . (2.27) stands for the Clapeyron equation of the thermodynamic systemof the d -dimensional RN-AdS black holes. III. ROTATING BLACK HOLESA. Thermodynamics
The AdS rotating black hole solution is given by the Kerr-AdS metric[20, 60, 61], ds = − ∆ ρ (cid:20) dt − a sin θ Ξ dφ (cid:21) + ρ ∆ dr + ρ Σ dθ + Σ sin θρ (cid:20) adt − r + a Ξ dφ (cid:21) + r cos θd Ω d − , (3.1)where ∆ = ( r + a ) (cid:18) r l (cid:19) − mr − d , Σ = 1 − a l cos θ, Ξ = 1 − a l , ρ = r + a cos θ, (3.2)and d Ω d denotes the metric element on a d -dimensional sphere. The associated thermody-namic quantities are M = ω d − π m Ξ (cid:18) d − (cid:19) , J = ω d − π ma Ξ , Ω = a l r + l r + a ,S = ω d − a + r ) r d − Ξ = A , V = r + Ad − (cid:20) a Ξ 1 + r /l ( d − r (cid:21) ,T = 12 π (cid:20) r + (cid:18) r l (cid:19) (cid:18) a + r + d − r (cid:19) − r + (cid:21) , (3.3)1in which r + meets ∆( r + ) = 0, that is ∆ = ( r + a ) (cid:16) r l (cid:17) − mr − d + = 0. Then4 πM Ξ ω d − (cid:0) d − Ξ (cid:1) = m = 12 r d − ( r + a ) (cid:18) r l (cid:19) , (3.4)for r + in terms of M and l . While expressing a = εl , we expanded[20] r + = k X I =0 r I ε I (3.5)to some given order k and solve Eq. (3.4) order by order. The first term yields the relation M = ω d − ( d − r d − π (cid:18) r l (cid:19) . (3.6)Using Eqs.(2.6) and (3.3), the equation of P = P ( v, T, J ) can be obtained P = Tv − ( d − π ( d − v + π ( d − d J ω d − ( d − d − v d − + o ( ε ) , (3.7)where v = r k + X I =1 v I ε I . (3.8)While J = 1 the isotherms are plotted in P − v diagrams at d = 4 , ,
10 respectively, whichare shown in Fig.5. The situations of negative pressure and the thermodynamics unstableregion with ∂P / ∂v > T < T c , where T c is the critical temperature of the Kerr-AdS black hole, unstablethermodynamic state appears, and while T < ˜ T , negative pressure emerges. The expressionof ˜ T can be derived from (3.7) as in (2.12), and the corresponding ˜ v is obtained by the way.˜ T = 2( d − π (2 d − v , ˜ v d − = π (2 d − d − d J ω d − ( d − d − d − . (3.9) B. Maxwell equal area law
We use Maxwell equal area law to simulate a possible phase transition process of the d -dimensional Kerr-AdS black hole as a thermodynamic system. We also take v and v as the specific volumes of the two associated phases in the phase transition respectively,and P as the constant pressure throughout the simulated process of the phase transition atconstant temperature T ( T ≤ T c ), which can be derived by Maxwell equal area law.2 v - - P (a) d = 4; J = 1 v - - P (b) d = 8; J = 1 v - P (c) d = 10; J = 1 FIG. 5: the isotherms in P − v diagrams . The four curves in each diagram correspond to T > T c , T = T c , T = ˜ T , T < ˜ T respectively. P ( v − v ) = v Z v P dv, (3.10)which can be rewritten as P ( v − v ) = T ln (cid:18) v v (cid:19) + ( d − π ( d − (cid:18) v − v (cid:19) − π ( d − d J d − ω d − ( d − d − (cid:18) v d − − v d − (cid:19) . (3.11)Combining with P = T v − ( d − π ( d − v + π ( d − d J ω d − ( d − d − v d − ,P = T v − ( d − π ( d − v + π ( d − d J ω d − ( d − d − v d − , (3.12)and setting x = v v , one can get T = ( d − π ( d −
2) (1 + x ) v x − π ( d − d J ω d − ( d − d − − x d − v d − x d − (1 − x ) ,P = ( d − π ( d −
2) 1 v x − π ( d − d J ω d − ( d − d − − x d − v d − x d − (1 − x ) ,v d − = J y ( x, ) y ( x ) , (3.13)where y ( x ) = π ( d − d ω d − ( d − d − (cid:18) (1 − x d − ) ln x + (cid:20) (1 − x d − )2 d − − x d − (cid:21) (1 − x ) (cid:19) , y ( x ) = 2( d − π ( d − x d − (1 − x ) ((1 + x ) ln x + 2(1 − x )) . (3.14)while x →
1, critical thermodynamics quantities can be got (3.13), v c = 4 d − (cid:20) π (2 d − d − J ( d − d − ω d − (cid:21) / (2 d − ,T c = 4( d − π (2 d − v c , P c = d − π ( d − v c . (3.15)The results are consistent with that in Ref.[20]. As T = χT c and χ ≤
1, we can derive from(3.13) and (3.15), χ ( d − x d − π (2 d − (cid:18) ( d − d − ω d − π (2 d − d − (cid:19) / (2 d − (cid:18) y ( x ) y ( x ) (cid:19) (2 d − / (2 d − = (1 + x ) π ( d − x d − (cid:18) y ( x ) y ( x ) (cid:19) − π ( d − d ω d − ( d − d − ( d −
3) (1 − x d − )1 − x . (3.16)So we can solve x at certain χ ( T ), then from (3.13) we can obtain the v , P and v = xv . Take J = 1, d = 4; 8; 10 respectively. In Fig.6, we plot the isotherms (blue curves) fordifferent χ in P − v diagrams and the isobars (red solid lines) which represent the simulatedphase transition processes derived from Maxwell equal area law. Like that in RN-AdS blackhole the processes become shorter with increasing temperatures until it turns into a point ata certain temperature, which is the critical temperature, and the point is the critical point ofthe black hole thermodynamic system. Every connected solid curve represents a processes ofstate change and phase transition, where no negative pressure and thermodynamic unstablestate appear. The boundary of the two-phase coexistence region is delineated by the dot-dashed line.In Table 2, we calculate x , v , P as χ = 0 . , . , . , . , . J = 0 . , . ,
1, and d = 4 , ,
10 respectively to find the influence of these parameters on the simulated phasetransition process and the two-phase coexistence region.From Table 2, one can see that x is unrelated to J , but incremental with the increase of χ / d at certain d / χ . v decreases with increasing χ / d and increases with the incremental J when the other parameters are fixed respectively. Similarly P increases with the incremental χ / d and decreases with increasing J while the other parameters are given respectively.4 v - - P FIG. 6:
The simulated phase transition(red solid lines) and the boundary of two phase coexistence(green dot-dashed curve) on the base of isotherms in P − v diagram for Kerr-AdS black hole with d = 8 , J = 5 . C. the relation of phase transition pressure to temperature
From (3.12) and T = χT c , we get P J / ( d − (cid:18) y ( x ) y ( x ) (cid:19) ( d − / ( d − = χ ( d − d − π (2 d − (cid:18) y ( x ) y ( x ) (cid:19) (2 d − / (2 d − (cid:20) ( d − d − ω d − π (2 d − d − (cid:21) / (2 d − − ( d − π ( d − (cid:18) y ( x ) y ( x ) (cid:19) + π ( d − d ω d − ( d − d − . (3.17)Combining (3.16), (3.17) and T = χT c , we plot the P J / ( d − − χ diagram and P − T diagrams in Fig.7 and Fig.8.From (3.13), we get P = y ( x ) , T = y ( x ) , (3.18)where y ( x ) = J − d − (cid:18) y ( x ) y ( x ) (cid:19) d − d − (cid:18) ( d − π ( d − x y ( x ) y ( x ) − π ( d − d ω d − ( d − d − − x d − x d − (1 − x ) (cid:19) ,y ( x ) = J − d − (cid:18) y ( x ) y ( x ) (cid:19) d − d − (cid:18) ( d − π ( d −
2) (1 + x ) x y ( x ) y ( x ) − π ( d − d ω d − ( d − d − − x d − x d − (1 − x ) (cid:19) . Then dP dT = y ′ ( x ) y ′ ( x ) , (3.19)5 TABLE II:
The parameters of two phase coexistence for Kerr-AdS black hole d = 4 d = 8 d = 10 J χ x v P x v P x v P . . which is the Clapeyron equation of the Kerr-AdS black hole.We can see from Fig.7, P ≥
0, and P → χ → T → d -dimensional Kerr-AdS black hole, as that of the d -dimensional RN-AdS blackhole. IV. DISCUSSIONS
In this paper we take AdS black holes as thermodynamic systems and find the stateequations in some region yield negative pressure and thermodynamic unstable states with ∂P / ∂v >
0. Comparing with phase transition in a usual liquid-gas system and using Maxwell6 Χ P J - d = d = d = FIG. 7: P J / ( d − − χ phase diagram for d -dimensional Kerr-AdS black hole with d = 4 , , . T P J = = = d = (a) d = 4; J = 1 T P J = = = d = (b) d = 8; J = 1 T P J = = = d = (c) d = 10; J = 1 FIG. 8: P − T phase diagrams at fixed J and d for d -dimensional Kerr-AdS black hole. equal area law, we simulate the phase transition process in AdS black holes. From Fig.2 andFig.6, one can see that, for both the d -dimensional RN-AdS black hole and the d -dimensionalKerr-AdS black hole, as T < T c a part of each isotherm could be replaced with an isobar,which means phase transition occurs and two phases coexist in the section. The simulatedphase transition belongs to the first order phase transition, and as T → T c the phasetransition process shortens gradually until it become a point at T = T c , where the phasetransition is the second order phase transition. The dimension d and the electric charge q (the angular momentum J ) also have effect on the simulated phase transition process. The7larger the dimension d , the greater the change of specific volume before and after the phasetransition. The larger the parameter q ( J ), the lower the pressure P , in the simulated phasetransition.Due to lack of the knowledge of chemical potential, the P − T curve for two phasecoexistence state of an usual thermodynamic system are obtained by experiment. Howeverthe slope of the curves can be calculated from the Clapeyron equation, dPdT = LT ( v β − v α ) , (4.1)in which L = T ( s β − s α ), v α and v β are the molar volumes of phase α and phase β respec-tively, and the Clapeyron equation accords with experiment result, which directly verifiesits thermodynamic correctness.In a general thermodynamic system, the phase transition means the change of organiza-tion structure of substance (including the changes of position and the orientation of atom,ion, and electron) and a different phase appear. Phase transition is a physical process, andit does not involve the chemical reaction, so the chemical compositions do not change in theprocess.Appropriate theoretical interpretation to the phase structure of AdS black hole thermo-dynamic system can help to know more about black hole thermodynamic properties, such asentropy, temperature and heat capacity, and it is significant for improving a self-consistentblack hole thermodynamic theory. This paper displays the P − T diagrams for the two phasecoexistence state and the Clapeyron equations of the black holes, which provide theoreticalbasis for investigation of phase transition and phase structure of black holes and help toexplore the theory of gravity. Acknowledgments
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