Phase transition and entropy force between two horizons in (n+2)-dimensional de Sitter space
aa r X i v : . [ h e p - t h ] M a r Phase transition and entropy force between two horizons in(n+2)-dimensional de Sitter space
Yang Zhang a,b , Wen-qi Wang a , Yu-bo Ma a,b , and Jun Wang c ∗ a Department of Physics, Shanxi Datong University, Datong 037009, China b Institute of Theoretical Physics, Shanxi Datong University, Datong 037009, China c School of Physics and Astronomy, Yunnan University, Kunming 650091, P. R. China
In this paper, the effect of the space-time dimension on effective thermodynamicquantities in (n+2)-dimensional Reissoner-Nordstrom-de Sitter space has been stud-ied. Based on derived effective thermodynamic quantities, conditions for the phasetransition are obtained. The result shows that the accelerating cosmic expansioncan be attained by the entropy force arisen from the interaction between horizons ofblack holes and our universe, which provides a possible way to explain the physicalmechanism for the accelerating cosmic expansion.
I. INTRODUCTION
It is well known that the cosmic accelerated expansion indicates that our universe is aasymptotical de Sitter one. Moreover, due to the success of AdS / CFT, it prompts us tosearch for the similar dual relationships in de Sitter space. Therefore, the research of deSitter space is not only of interest to the theory itself, but also the need of the reality.In de Sitter space, the radiation temperature on the horizon of black holes and the universeis generally not the same. Therefore, the stability of the thermodynamic equilibrium cannot be protected in it, which makes troubles to corresponding researches. In recent years,study on thermodynamic properties of de Sitter space is getting more and more attention[1–12]. In the inflationary period, our universe seem to be a quasi de Sitter space, in whichthe cosmological constant is introduced as the vacuum energy, which is a candidate for darkenergy. If the cosmological constant corresponds to dark energy, our universe will goes intoa new phase in de Sitter space. In order to construct the entire evolutionary history of ouruniverse, and understand the intrinsic reason for the cosmic accelerated expansion, both the ∗ E-mail: [email protected] classic and quantum nature of de Sitter space should be studied.For a multi-horizon de Sitter space, although different horizons have different tempera-tures, thermodynamic quantities on horizons of black holes and the universe are functionsdepended on variables of mass, electric charge, cosmological constant and so on. Form thispoint of view, thermodynamic quantities on horizons are not individual. Based on this fact,effective thermodynamic quantities can be introduced. Considering the correlation betweenhorizons of black holes and the universe, we have studied the phase transition and the crit-ical phenomenon in RN-dS black holes with four-dimension and high-dimension by usingeffective thermodynamic quantities, respectively. Moreover, the entropy for the interactionbetween horizons of black holes and the universe is also obtained [13–17]. When we considerthe cosmological constant as a thermodynamic state parameter with the thermodynamicpressure, the result shows that de Sitter space not only has a critical behavior similar to thevan der Waals system [17, 18], but also take second-order phase transition similar to AdSblack hole [19–29]. However, first-order phase transition similar to AdS black hole is notexisted. In this work, we investigate the issue of the phase transition in a high-dimensionalde Sitter space, and analyze the effect of the dimension on the phase transition and theentropy produced by two interactive horizons.Nine years ago, Verlinde [30] proposed to link gravity with an entropic force. The ensuingconjecture was proved recently [31, 32], in a purely classical environment and then extendedto a quantal bosonic system in Ref. [31]. In 1998, the result of the observational data fromthe type Ia supernovae (SNe Ia) [40, 41] indicates that our universe presently experiencesan accelerating expansion, which contrasts to the one given in general relativity (GR) byAlbert Einstein. In order to explain this observational phenomenon, a variety of proposalhave been proposed. The theory of “early dark energy” proposed by Adam Riess [33, 34] isone of them, where dark energy [42, 43] as an exotic component with large negative pressureseems to be the cause of this observational phenomenon. According to the observations,dark energy occupies about 73% in cosmic components. Therefore, one believe that thepresent accelerating expansion of our universe should be caused by dark energy. Then a lotof dark energy models have been proposed. However, up to now, the nature of dark energyis not clear.Based on the entropy caused by the interaction between horizons of black holes and theuniverse, the relationship between the entropy force and the position ratio of the two hori-zons is obtained. When the position ratio of the black hole horizon to the universe horizonis greater (less) than a certain value, the entropy force between the two horizons is repul-sive (attractive), which indicates that the expansion of the universe horizon is accelerating(decelerating). While when it equal to the certain value, the entropy force is absent, andthen the expansion of the universe horizon is uniform. According to this, we suppose thatthe entropy force between the two horizons can be seen as a candidate to cause the cosmicaccelerated expansion.This paper is organized as follows. According to Refs. [16–18], a briefly review for theeffective thermodynamic quantities, the conditions for the phase transition and the effectof the dimension on the phase transition in ( n + 2) − dimensional Reissoner-Nordstrom-deSitter (DRNdS) space is given in the next section. In section 3, the entropy force of theinteraction between horizons of black holes and the universe is derived, and then the effectof the dimension on it is explored. Moreover, the relationship between the entropy force andthe position ratio of the two horizons is obtained. Conclusions and discussions are given inthe last section. The units G = ~ = k B = c = 1 are used throughout this work. II. EFFECTIVE THERMODYNAMIC QUANTITIES
The metric of ( n + 2) − dimensional DRNdS space is [35]: ds = − f ( r ) dt + f − ( r ) dr + r d Ω n (2.1)where the metric function is f ( r ) = 1 − ω n Mr n − + nω n Q n − r n − − r l , ω n = 16 πGnV ol ( S n ) . (2.2)Here G is the gravitational constant in n + 2 − dimensional space, l is the curvature radius ofdS space, V ol ( S n ) denotes the volume of a unit n − sphere d Ω n , M is an integration constantand Q is the electric/magnetic charge of Maxwell field.In n + 2 − dimensional DRNdS space, positions of the black hole horizon r + and theuniverse horizon r c can be determined when f ( r + ,c ) = 0. Moreover, thermodynamic quan-tities on these two horizons satisfy the first law of thermodynamics, respectively [3, 5, 35].However, thermodynamic systems denoted by the two horizons are not independent, sincethermodynamic quantities on them are functions depended on variables of mass M , electric - - - C p eff n = = = FIG. 1: (color online). C P eff − x diagram for Q = 0 . r c = 1 and n = 2; 4; 6,respectively. - - α n = = = FIG. 2: (color online). α − x diagram for Q = 0 . r c = 1 and n = 2; 4; 6,respectively. charge Q and cosmological constant l satisfy the first law of thermodynamics. When pa-rameters of state of n +2 − dimensional DRNdS space satisfy the first law of thermodynamics,the entropy is [16–18] S = V ol ( S n )4 G r nc (1 + x n + f n ( x )) = S c, + + S AB , (2.3)where x = r + /r c , S c, + = V ol ( S n )4 G r nc (1 + x n ) and S AB = V ol ( S n )4 G r nc f n ( x ) are entropies with andwithout the interaction between the two horizons, respectively, and f n ( x ) = 3 n + 22 n + 1 (1 − x n +1 ) n/ ( n +1) − ( n + 1)(1 + x n +1 ) + (2 n + 1)(1 − x n +1 − x n +1 )(2 n + 1)(1 − x n +1 ) . (2.4)The volume of n + 2 − dimensionalDRNdS space is[3, 7, 13] V = V c − V + = V ol ( S n )( n + 1) r n +1 c (1 − x n +1 ) . (2.5) - - κ T eff n = = = FIG. 3: (color online). κ T eff − x diagram for Q = 0 . r c = 1 and n = 2; 4; 6,respectively. When parameters of state of n + 2 − dimensional DRNdS space satisfy the first law ofthermodynamics, the effective temperature is [16–18] T eff = (1 − x n +1 ) ( ∂M/∂x ) r c (1 − x n +1 ) + r c x n ( ∂M/∂r c ) x V ol ( S n ) r nc x n − (1 + x n +2 )= B ( x ) V ol ( S n ) r c x n − ω n (1 + x n +2 ) , (2.6)where B ( x ) = x n [( n − x n − − ( n + 1) x n + 2 x n − + ( n − x n − (1 − x )] − nω n Q [( n − x n +1 (1 − x n ) − nx n +1 + ( n −
1) + ( n + 1) x n ]8( n − r n − c = x n [( n − x n − − ( n + 1) x n + 2 x n − + ( n − x n − (1 − x )] − φ c ( n − n − x n +1 (1 − x n ) − nx n +1 + ( n −
1) + ( n + 1) x n ] n , (2.7)where φ c = n n − ω n Qr n − c is electric potential on the universe horizon. The effective pressure P eff , isochoric heat capacity C veff and isobaric heat capacity C P veff in n + 2 − dimensionalDRNdS spaceare P eff = D ( x ) ω n V ol ( S n )(1 − x n +1 ) r c x n − (1 + x n +2 ) , (2.8)where D ( x ) = (cid:20) ( n − x n − − ( n + 1) x n + 2 x n − − nω n Q (2 nx n +1 − ( n − − ( n + 1) x n )8( n − r n − c x n (cid:21) × (1 + x n + f ( x )) (2.9) − (cid:20) ( n − x n − (1 − x ) − nω n Q (1 − x n )8 r n − c x n − (cid:21) (cid:18) x n − + f ′ ( x ) n (cid:19) (1 − x n +1 ) ,C V = T eff (cid:18) ∂S∂T eff (cid:19) V = T eff (cid:16) ∂S∂r c (cid:17) x (cid:0) ∂V∂x (cid:1) r c − (cid:0) ∂S∂x (cid:1) r c (cid:16) ∂V∂r c (cid:17) x (cid:0) ∂V∂x (cid:1) r c (cid:16) ∂T eff ∂r c (cid:17) x − (cid:16) ∂V∂r c (cid:17) x (cid:16) ∂T eff ∂x (cid:17) r c (2.10)= 14 G (1 − x n +1 ) × − V ol ( S n ) r nc B ( x ) nx n (1 + x n +2 ) ¯ B ( x ) x n +1 (1 + x n +2 ) − (1 − x n +1 ) x (1 + x n +2 ) B ′ ( x ) − B ( x )[2 n − n + 1) x n +2 ]where¯ B ( x ) = x n [( n − x n − − ( n + 1) x n + 2 x n − + ( n − x n − (1 − x )] − nω n Q (2 n − n − x n +1 (1 − x n ) − nx n +1 + ( n −
1) + ( n + 1) x n ]8( n − r n − c ,B ′ ( x ) = dB ( x ) dx , D ′ ( x ) = dD ( x ) dx , (2.11) C P eff = T eff (cid:18) ∂S∂T eff (cid:19) P eff = T eff (cid:16) ∂S∂r c (cid:17) x (cid:16) ∂P eff ∂x (cid:17) r c − (cid:0) ∂S∂x (cid:1) r c (cid:16) ∂P eff ∂r c (cid:17) x (cid:16) ∂P eff ∂x (cid:17) r c (cid:16) ∂T eff ∂r c (cid:17) x − (cid:16) ∂P eff ∂r c (cid:17) x (cid:16) ∂T eff ∂x (cid:17) r c = r nc V ol ( S n ) B ( x ) E ( x )4 GH ( x ) , (2.12)where E ( x ) = [ nx n − + f ′ ( x )][ ¯ D ( x ) − D ( x )](1 − x n +1 ) x (1 + x n +2 ) − n [1 + x n + f ( x )] { D ′ ( x ) x (1 − x n +1 )(1 + x n +2 ) − D ( x )[( n − − nx n +1 + (2 n + 1) x n +2 − (3 n + 2) x n +3 ] } ,H ( x ) = ¯ B ( x ) { D ′ ( x ) x (1 − x n +1 )(1 + x n +2 ) − D ( x )[( n − − nx n +1 + (2 n + 1) x n +2 − (3 n + 2) x n +3 ] } (2.13)+(1 − x n +1 )[ ¯ D ( x ) − D ( x )] (cid:2) x (1 + x n +2 ) B ′ ( x ) − B ( x )[2 n − n + 1) x n +2 ] (cid:3) . ¯ D ( x ) = nω n Q (2 nx n +1 − ( n − − ( n + 1) x n )4 r n − c x n (1 + x n + f ( x )) − n ( n − ω n Q (1 − x n )4 r n − c x n − (cid:18) x n − + f ′ ( x ) n (cid:19) (1 − x n +1 ) . The coefficient of isobaric volume expansion and isothermal compressibility in n + 2 − di-mensional DRNdS spaceis given by α = 1 V (cid:18) ∂V∂T eff (cid:19) P eff = 1 V (cid:16) ∂V∂r c (cid:17) x (cid:16) ∂P eff ∂x (cid:17) r c − (cid:0) ∂V∂x (cid:1) r c (cid:16) ∂P eff ∂r c (cid:17) x (cid:16) ∂P eff ∂x (cid:17) r c (cid:16) ∂T eff ∂r c (cid:17) x − (cid:16) ∂P eff ∂r c (cid:17) x (cid:16) ∂T eff ∂x (cid:17) r c = − ω n ( n + 1) V ol ( S n ) x n − (1 + x n +2 ) H ( x ) r c { x n +1 [ ¯ D ( x ) − D ( x )](1 + x n +2 ) (2.14)+ D ′ ( x ) x (1 − x n +1 )(1 + x n +2 ) − D ( x )[( n − − nx n +1 + (2 n + 1) x n +2 − (3 n + 2) x n +3 ] } .κ T eff = − V (cid:18) ∂V∂P eff (cid:19) T eff = 1 V (cid:16) ∂V∂r c (cid:17) x (cid:16) ∂T eff ∂x (cid:17) r c − (cid:0) ∂V∂x (cid:1) r c (cid:16) ∂T eff ∂r c (cid:17) x (cid:16) ∂P eff ∂x (cid:17) r c (cid:16) ∂T eff ∂r c (cid:17) x − (cid:16) ∂P eff ∂r c (cid:17) x (cid:16) ∂T eff ∂x (cid:17) r c = r c ω n ( n + 1) V ol ( S n )(1 − x n +1 ) x n − (1 + x n +2 ) H ( x ) × (2.15) (cid:8) (1 − x n +1 ) (cid:2) x (1 + x n +2 ) B ′ ( x ) − B ( x )[2 n − n + 1) x n +2 ] (cid:3) − x n +1 (1 + x n +2 ) ¯ B ( x ) (cid:9) Numerical solutions for the isobaric heat capacity C p eff and coefficients of isobaric volumeexpansion α and isothermal compressibility κ T eff with the position ratio of the black hole TABLE I:
Critical values of the effective thermodynamic system for different nn = 2 n = 4 n = 6 x c T ceff P ceff horizon to the universe horizon x have been given in Fig. 1, Fig. 2 and Fig. 3, respectively.Form the figures, it is clear that values of C p eff , α and κ T eff have sudden change withthe charge of the spacetime is a constant, which is similar to the Van der Waals system.Moreover, as the dimension of the space increases, the value of x to denote the suddenchange also increases. This indicates that the point of the phase transition is closely relatedto the dimensions of the space time.From Table 1, it is clear that the phase transition point is different with different dimen-sions. Moreover, as the dimension increases, the critical value of the phase transition pointand the effective pressure and temperature are all increased. III. ENTROPY FORCE
The entropy force of a thermodynamic system can be expressed as [30–32, 36–39] F = − T ∂S∂r , (3.1)where T is the temperature and γ is the radius.From Eq.(2.3), the entropy caused by the interaction between horizons of black holes andthe universe is S AB = V ol ( S n )4 G r nc f n ( x ) . (3.2)From Fig.4, it shows that as the dimension increases, the intersectional point of the curveand the x − axis is moving to the right.In other words, the value of x increases with thedimension,which denotes the point where the entropy caused by the interaction betweenhorizons of black holes and the universe changes between positive and negative values. Theentropy given in Eq. (2.4) does not contain explicit electric charge Q dependent Q terms. - f n ( x ) n = = = FIG. 4: (color online). f n ( x ) − x diagram for V ol ( S n )4 G =1 , r nc = 1 and n = 2; 4; 6,respectively. - - - - ( x ) FIG. 5: (color online). F ( x ) − x diagram for Q = 0 . r c = 1 and n = 2; 4; 6,respectively. From Eq. (3.1), the entropy force of the two interactive horizons can be given as F = − T eff (cid:18) ∂S AB ∂r (cid:19) T eff , (3.3)where T eff is the effective temperature of the considering case and r = r c − r + = r c (1 − x ).Then it gives F ( x ) = − T eff (cid:16) ∂S f ∂r c (cid:17) x (cid:16) ∂T eff ∂x (cid:17) r c − (cid:16) ∂S f ∂x (cid:17) r c (cid:16) ∂T eff ∂r c (cid:17) x (1 − x ) (cid:16) ∂T eff ∂x (cid:17) r c + r c (cid:16) ∂T eff ∂r c (cid:17) x (3.4)= − B ( x ) r n − c Gx n − ω n (1 + x n +2 ) × nf n ( x ) [ x (1 + x n +2 ) B ′ ( x ) − B ( x )[2 n − n + 1) x n +2 ]] + x (1 + x n +2 ) ¯ B ( x ) f ′ n ( x )(1 − x ) [ x (1 + x n +2 ) B ′ ( x ) − B ( x )[2 n − n + 1) x n +2 ]] + x ¯ B ( x )(1 + x n +2 ) . Fig.5 shows that the entropy force increases with the dimension. Moreover, when n = 2and x = x = 0 . n = 4 and x = x = 0 . n = 6 and x = x = 0 . F ( x ) = 0, respectively. It indicates that the value of x increases with the dimension,which denotes the point where the direction of the entropy force changes.Fig. 6 shows that when Q = 0 .
001 and x = x = 0 . , Q = 0 .
01 and x = x = 0 . Q = 0 . x = x = 0 . , F ( x ) = 0 respectively. It implies that as the electric F ( x ) Q = (cid:0) = (cid:1) = FIG. 6: (color online). F ( x ) − x diagram for n = 2, r c = 1 and Q = 0 . .
01; 0 . charge increases, the value of x decreases, which denotes the point where the entropy forcechanges between positive and negative values.From Fig. 5, we can obtain that when x → , F ( x → → ∞ , and then according to Eq.(2.6), T eff →
0. This result indicates that the interaction between horizons of black holesand the universe tends to infinity, which contrast to the third law of thermodynamics. Inorder to protect the laws of thermodynamics, the black hole horizon and the cosmologicalhorizon can not coincide with each other. Based on this fact, we take 1 − ∆ x as the maximumvalue of x , where ∆ x is a minor dimensionless quantity. The value of ∆ x can be determinedby the speed of the cosmic accelerated expansion at the position x .According to the expression of the entropy force, when x < x < − ∆ x, F ( x ) > x < x < − ∆ x . While when 0 < x < x , F ( x ) <
0, which indicates thatthe interaction between horizons of black holes and the universe is attractive, and then theexpansion of the universe is variable deceleration in this interval.From Fig. 5, we find that when the area enclosed by the curve F ( x ) − x and the x − axiswith the interval of x < x < − ∆ x is larger than the area enclosed by the same curveand the x − axis with the interval of 0 < x < x ,the cosmic expansion is from accelerationto deceleration. It gives an expanding universe. While when the former area is less than orequal to the latter one, the cosmic expansion is from acceleration to deceleration. Moreover,when these two areas are equal at the position ratio x , which belongs to the interval of¯ x < x < x , the universe is accelerated shrinkage from the position ratio ¯ x to the positionratio x , where ¯ x is determined when the area between the curve and the x-axis with the0interval of [¯ x, − ∆ x ] is zero. After the universeshrink to the position ratio x = 1 − ∆ x, theevolution of the universe begins the next cycle. It gives a oscillating universe. IV. CONCLUSIONS
When horizons of black holes and the universe are irrelevant, thermodynamic systemsof them are independent. Since the radiational temperature on them is different, the re-quirement of thermodynamic equilibrium stability can not be meet. Therefore, the spaceis unstable. While when they are related, the effective temperature T eff and pressure P eff for DRNdS space can be obtained from Eqs.(2.6) and (2.8). According to curves C P eff − x , α − x , and κ T eff − x , when x = x c , the phase transition of DRNdS space time occurs.Since its entropy and volume are continuous, the phase transition is the second-order oneaccording to Ehrenfest’s classification. It is similar to the case occured in AdS black holes[19–24, 44, 45]. From Eq. (2.10), we find that the isochoric heat capacity C v of DRNdS spaceis non-trivial, which is similar to the system of Van der Waals, but different from AdS blackholes. In second 2, the effect of the dimension on the phase transition point is analyzed,which lays the foundation for the further study of the thermodynamic characteristics of thehigh-dimensional complex dS space.From Fig. 5, we find that when the area enclosed by the curve F ( x ) − x and the x − axiswith the interval of x < x < − ∆ x is larger than the area enclosed by the same curveand the x − axis with the interval of 0 < x < x , the cosmic expansion is from accelerationto deceleration. It gives an expanding universe. While when the former area is less than orequal to the latter one, the cosmic expansion is from acceleration to deceleration. Moreover,when these two areas are equal at the position ratio x , which belongs to the interval of¯ x < x < x , the universe is accelerated shrinkage from the position ratio ¯ x to the positionratio x , where ¯ x is determined when the area between the curve and the x − axis with theinterval of [¯ x, − ∆ x ] is zero. After the universe shrink to the position ratio x = 1 − ∆ x ,the evolution of the universe begins the next cycle. It gives a oscillating universe.Whether the universe is an expanding one or a oscillating one is determined by the valueof the minor dimensionless quantity. From Fig. 5 and Fig.6, we find that the position, wherethe entropy force changes between positive and negative values, is greatly affected by thedimension, but commonly by the electric charge. Therefore, the effect of the dimension on1the cosmic expansion is greater than the electric charge. Moreover, since the curve F ( x ) − x is continuous at the phase transition point x c , the entropy force can not be affected by thephase transition in the space with a given dimension and electric charge. The amplitudeand the value of the entropy force is only determined by the position ratio x . According toour research result, the entropy force between horizons of black holes and the universe canbe taken as one of the reasons for the cosmic expansion, which provides a new approach forpeople to explore the physical mechanism of the cosmic expansion. Acknowledgements
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