Thermodynamics of de Sitter black hole in massive gravity
aa r X i v : . [ h e p - t h ] A ug EPJ manuscript No. (will be inserted by the editor)
Thermodynamics of de Sitter black holes in massive gravity
Yu-Bo Ma , Ren Zhao , Shuo Cao School of Physics, Shanxi Datong University, Datong, 037009, China Institute of Theoretical Physics, Shanxi Datong University, Datong 037009, China Department of Astronomy, Beijing Normal University, Beijing, 100875, Chinathe date of receipt and acceptance should be inserted later
Abstract.
In this paper, by taking de Sitter space-time as a thermodynamic system, we study the effectivethermodynamic quantities of de Sitter black holes in massive gravity, and furthermore obtain the effectivethermodynamic quantities of the space-time. Our results show that the entropy of this type of space-timetakes the same form as that in Reissner-Nordstr¨om-de Sitter space-time, which lays a solid foundation fordeeply understanding the universal thermodynamic characteristics of de Sitter space-time in the future.Moreover, our analysis indicates that the effective thermodynamic quantities and relevant parameters playa very important role in the investigation of the stability and evolution of de Sitter space-time.
PACS.
The study of the thermodynamic characteristics of deSitter space-time has arouse extensive attention in the re-cent years [1–21]. At the stage of cosmological inflation inthe early time, our universe behaves like a quasi-de Sit-ter space-time, in which the cosmological constant takesthe form of vacuum energy. Moreover, if the dark energyis simply a cosmological constant, i.e., a component withconstant equation of state, our universe will evolve into anew stage of de Sitter space-time in this simplest scenario.Therefore, a better knowledge of de Sitter space-time (es-pecially its classical and quantum characteristics) is veryimportant to construct the general framework of cosmicevolution. In the previous works, the black hole horizonand the cosmological horizon are always treated as two in-dependent thermodynamic systems [4–7, 13], from whichthe thermodynamic volume of de Sitter space-time as wellas the corresponding thermodynamic quantities satisfyingthe first thermodynamics law were obtained [3]. It is com-monly recognized that the entropy of de Sitter space-timeis the sum of that for the two types of horizons [7, 14],however, such statement concerning the nature of de Sitterspace-time entropy still needs to be checked with adequatephysical explanation.Considering the fact that all thermodynamic quanti-ties related to the black hole horizon and the cosmologicalhorizon in de Sitter space-time can be expressed as a func-tion of mass M , electric charge Q , and cosmological con-stant Λ , it is natural to consider the dependency betweenthe two types of thermodynamic quantities. More specifi-cally, the discussion of the following two problems is very a e-mail: [email protected] significant to study the stability and evolution of de Sitterspace-time: Do the thermodynamic quantities follow thebehavior of their counterparts in AdS black holes, espe-cially when the black hole horizon is correlated with thatof the cosmological horizon? What is the specific relationbetween the entropy of de Sitter space-time and that ofthe two horizons (the black hole horizon and the cosmo-logical horizon)? The above two problems also provide themain motivation of this paper.Following this direction, in our analysis we obtain theeffective thermodynamic quantities of de Sitter black holesin massive gravity (DSBHMG), based on the correlationbetween the black hole horizon and the cosmological hori-zon. Our results show that the entropy of this type ofspace-time takes the same form as that in Reissner-Nordstromde Sitter space-time, which lays a solid foundation fordeeply understanding the universal thermodynamic char-acteristics of de Sitter space-time in the future. This paperis organized as follows. In Sec.II, we briefly introduce thethermodynamic quantities of the horizons of black holesand the Universe in DSBHMG, and furthermore obtainthe electric charge Q when the two horizons show thesame radioactive temperature. In Sec.III, taking the corre-lation between the two horizons into consideration, we willpresent the equivalent thermodynamic quantities of DS-BHMG satisfying the first thermodynamic law, and per-form a quantitative analysis of the corresponding effectivetemperature and pressure. Finally, the main conclusionsare summarized in Sec.IV. Throughout the paper we usethe units G = ¯ h = k B = c = 1 . Yu-Bo Ma, Ren Zhao, Shuo Cao: Thermodynamics of de Sitter black holes in massive gravity
In the framework of (3 + 1)-dimensional massive grav-ity with a Maxwell field (denoting F µν as the Maxwellfield-strength tensor), the corresponding action always ex-presses as [22–25], S = 12 k Z d x √ g " R − − F + m X i c i µ i , (1)Here k = +1 , , − µ i represents the contribution of the ma-trix √ g µα f αν with fixed symmetric tensor f µν . Therefore,generated form the above action, the space-time metricof static black holes (denoting h ij as Einstein space withconstant curvature) can be written as ds = − f ( r ) dt + f − dr + r h ij dx i dx j , i, j = 1 , f ( r ) = k − Λ3 r − m r + q r + c m r + m c (3)Note that the positions of black hole horizon r + and cos-mic horizon r c are determined when f ( r + ,C ) = 0. Fig. 1.
The metric function f ( r ) varying with r . In Fig. 1 we display the behavior of the metric function f ( r ), where the parameters are chosen as Λ = 1 , m =30 , m = 2 . , c = 2 , c = 3 . , q = 1 .
7, while k is fixed at1 ,
0, and −
1. It is obvious that there are two intersectionpoints between f ( r ) and the axis of r , which respectivelycorrespond to the positions of black hole horizon r + andcosmological horizon r c . Thus, the mass m can be ex-pressed in terms of r + ,c as M = m k + m c ) r c x (1 + x )2(1 + x + x )+ q (1 + x )(1 + x )8 r c x (1 + x + x ) + r c m c x x + x ) (4) andΛ3 r c (1 + x + x ) = k − q r c x + c m r c (1 + x ) + m c (5)where x = r + /r c . The temperature of the black holehorizons and cosmic horizon can be written as [28] T + ,c = ± f ( r + ,c )4 π = 14 πr + ,c (cid:20) k − Λ r ,c − q r ,c + m c r + ,c + m c (cid:21) (6)Turning to the contribution of the electrical charge q , itwill also generate a chemical potential as µ + ,c = qr + ,c . (7)According to the Hamiltonian approach, we have the mass M and electric charge Q as M = ν m π , Q = ν q π . (8)and the entropy of the two horizons respectively expressas S + ,c = ν r ,c (9)where ν is the area of a unit volume of constant ( t, r )space (which equals to 4 π for k = 0). It is apparent thatthe thermodynamic quantities corresponding to the twohorizons satisfy the first law of thermodynamics dM = T + ,c dS + ,c + V + ,c dP + µ + ,c dQ (10)where V + ,c = ν r ,c , P = − Λ8 π (11)When the temperature of the black hole horizon is equalto that of the cosmological horizon, the electric charge Q and the cosmological constant Λ are related as1 r + [ k − Λ r − q r + m c r + + m c ]= − r c [ k − Λ r c − q r c + m c r c + m c ] (12)As can be seen from Eq. (5) and (12) the electric chargeof the system satisfies the following expression q (1 + x ) r c x = k + c m r c x x ) + m c . (13)When taking T + = T c , the combination of Eqs. (5), (6)and (13) will lead to the temperature T as T = T + = T c = (1 − x )2 πr c (1 + x ) (cid:20) k + m c r c (1 + 4 x + x )4(1 + x ) + m c (cid:21) (14) u-Bo Ma, Ren Zhao, Shuo Cao: Thermodynamics of de Sitter black holes in massive gravity 3 Considering the connection between the black hole hori-zon and the cosmological horizon, we can derive the effec-tive thermodynamic quantities and corresponding first lawof black hole thermodynamics as dM = T eff dS − P eff dV + φ eff dQ, (15)where the thermodynamic volume is defined by [3, 5, 6, 29] V = 4 π (cid:0) r c − r (cid:1) . (16)It is obvious that there exit three real roots for the equa-tion f ( r ) = 0: the cosmological horizon (CEH) r = r c ,the inner (Cauchy) horizon of black holes, and the outerhorizon (BEH) r = r + of black holes. Moreover, the deSitter space-time is characterized by Λ >
0, while Λ < S = πbr c (cid:2) x + f ( x ) (cid:3) , (17)Here the undefined function f ( x ) represents the extra con-tribution from the correlations of the two horizons. FromEq. (15), we can obtain the effective temperature T eff andpressure P eff T eff = (cid:18) ∂M∂S (cid:19) Q,V = (cid:0) ∂M∂x (cid:1) r c (cid:16) ∂V∂r c (cid:17) x − (cid:0) ∂V∂x (cid:1) r c (cid:16) ∂M∂r c (cid:17) x (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) ∂S∂r c (cid:17) x , (18) P eff = − (cid:18) ∂M∂V (cid:19) Q,S = − (cid:0) ∂M∂x (cid:1) r c (cid:16) ∂S∂r c (cid:17) x − (cid:0) ∂S∂x (cid:1) r c (cid:16) ∂M∂r c (cid:17) x (cid:0) ∂V∂x (cid:1) r c (cid:16) ∂S∂r c (cid:17) x − (cid:0) ∂S∂x (cid:1) r c (cid:16) ∂V∂r c (cid:17) x . (19)Combining Eqs.(4), (16) and (17), one can obtain T eff = B ( x, q )2 πr c [2 x (1 + x + f ( x )) + (1 − x )(2 x + f ′ ( x ))] , (20)where B ( x, q ) =( k + m c ) (1 + x − x + x + x )(1 + x + x ) − q r c (1 + x + x − x + x + x + x ) x (1 + x + x )+ r c m c x (2 + x )(1 − x ) + 2 x x + x ) . (21) P eff = D ( x, q )8 πr c [2 x (1 + x + f ( x )) + (1 − x )(2 x + f ′ ( x ))] , (22) D ( x, q ) = ( k + m c )(1 + x + x ) (cid:8)(cid:0) (1 + 2 x )(1 + x + f ( x )) (cid:1) − x (1 + x )(1 + x + x )(2 x + f ′ ( x )) (cid:9) − q r c x (1 + x + x ) (cid:8)(cid:0) x + 3 x ) (cid:1) (1 + x + f ( x )) − x (1 + x )(1 + x )(1 + x + x )(2 x + f ′ ( x ))) } + r c m c x (1 + x + x ) { (2(2 + x )(1 + x + f ( x )) − x (1 + x + x )(2 x + f ′ ( x )) (cid:1)(cid:9) , (23)When the temperature of the black hole horizon isequal to that of the cosmological horizon, the effectivetemperature of the space-time should be T eff = T = T c = T + . (24)Then substituting Eq. (14) into Eq. (20), we get(1 − x )(1 + x ) (cid:20) k + m c + m c r c (1 + 4 x + x )4(1 + x ) (cid:21) (1 + x + x )= B ( x )[2 x (1 + x + f ( x )) + (1 − x )(2 x + f ′ ( x ))] . (25)where B ( x ) = 2 x (1 + x )(1 + x ) (cid:20) ( k + m c ) + r c m c x ) (1 + 4 x + x ) (cid:21) . (26)Then Eq. (25) will transform into f ′ ( x ) + 2 x − x f ( x ) = 2 x (2 x + x − − x ) . (27)with the corresponding solution as f ( x ) = − (cid:0) − x − x (cid:1) − x ) + C (cid:0) − x (cid:1) / . (28)Considering the initial condition of f (0) = 0, we can ob-tain C = 8 / T eff = B ( x, q )(1 − x )4 πr c x (1 + x ) , P eff = D ( x, q )(1 − x )16 πr c x (1 + x ) . (29)Based on the above equations, the P eff − x and T eff − x diagrams could be derived by taking different value of k , q , m , c and c (when taking r c = 1). Yu-Bo Ma, Ren Zhao, Shuo Cao: Thermodynamics of de Sitter black holes in massive gravity
Fig. 2.
The P eff − x diagram when the parameter k is fixedat 1, 0 and -1, respectively. The other parameters are fixed at m = 2 . , c = 2 , c = 3 . , q = 1 . Fig. 3.
The P eff − x diagram varying with the parametersof m , c , c and q , while the other two parameters are fixed at k = − r c = 1. Fig. 4.
The S ( x ) − x and f ( x ) − x diagrams with r c = 1. Fig. 5.
The T eff − x diagram when k is fixed at 1, 0 and -1,respectively. The other parameters are fixed at m = 2 . , c =2 , c = 3 . , q = 1 . r c = 1. In Fig. 2 and 3, we illustrate an example of the P eff − x diagram with different value of relevant parameters, fromwhich one could clearly see the effect of these parameterson the effective pressure of RN-dSQ space-time. Followingthe same procedure by inserting Eq. (28) into Eq. (17), wecan also obtain the S ( x ) − x and f ( x ) − x diagrams with r c = 1, which are explicitly shown in Fig. 4. Similarly,in Fig. 5 and 6, we show the evolution of the T eff − x diagram with different value of relevant parameters, fromwhich one could perceive the effect of these parameters onthe effective temperature of RN-dSQ space-time. Fig. 6.
The T eff − x diagram varying with the parametersof m , c , c and q , while the other two parameters are fixed at k = − r c = 1. Moreover, it is shown that the change of these relatedparameters may also significantly affect the position of thestability and phase-transition points, which can be clearlyseen from the results presented in Table 1 and 2.
Parametric x c T ceff x k = 1 0.3374 1.7554 0.2173 k = 0 0.3468 1.5799 0.2243 k = − Table 1.
Summary of the highest effective temperature T ceff and the corresponding x c for different curves in Fig. 5. Thevalue of x when the effective temperature reaches zero is alsolisted. In this paper, by taking de Sitter space-time as a ther-modynamic system, we study the effective thermodynamicquantities of de Sitter black holes in massive gravity, and u-Bo Ma, Ren Zhao, Shuo Cao: Thermodynamics of de Sitter black holes in massive gravity 5Parametric x c T ceff x q = 1 . k = − q = 1 . q = 2 . m = 1 . k = − m = 2 .
12 0.35710 1.41210 0.22310 m = 3 .
12 0.24946 5.61153 0.15324 c = 1 . k = − c = 2 . c = 3 . c = 2 .
18 0.41792 0.75768 0.27926 k = − c = 3 .
18 0.35710 1.41210 0.23210 c = 4 .
18 0.31677 2.22455 0.20218
Table 2.
Summary of the highest effective temperature T ceff and the corresponding x c for different curves in Fig. 6. Thevalue of x when the effective temperature reaches zero is alsolisted. furthermore obtain the effective thermodynamic quanti-ties of the space-time. Here we summarize our main con-clusions in more detail: – In the previous analysis without considering the corre-lation between the black hole horizon and the cosmo-logical horizon, i.e., the two horizons are always treatedas independent thermodynamic systems with differenttemperature, the space-time does not satisfy the re-quirement of thermodynamic stability. In this paper,we find that the establishment of the correlation be-tween the two horizons will generate the common effec-tive temperature T eff , which may represent the mosttypical thermodynamic feature of RN-dSQ space-time. – As can be clearly seen from the S ( x ) − x and T eff − x diagrams, RN-dSQ space-time in unstable under thecondition of x > x c and x < x . This result indicatesthat there exist only the RN-dSQ black holes satisfyingthe condition of x < x < x c , which lays a solidtheoretical foundation for the search of black holes inthe Universe. – We find that the interaction term f ( x ) in the entropyof RN-dSQ space-time takes the same form of that inRN-dS space-time. Considering that the entropy in thetwo types of space-time is the function of the positionof the horizon, which has no relation with other param-eters including the electric charge ( Q ) and the constant(Λ), the entropy in the two types of space-time shouldtake the same form. This finding may contribute tothe deep understanding the universal thermodynamiccharacteristics of de Sitter space-time in the future. Acknowledgments
The authors declare that there is no conflict of in-terest regarding the publication of this paper. This workwas supported by the Young Scientists Fund of the Na-tional Natural Science Foundation of China (Grant Nos.11605107 and 11503001), in part by the National Natural Science Foundation of China (Grant No. 11475108), Sup-ported by Program for the Innovative Talents of HigherLearning Institutions of Shanxi, the Natural Science Foun-dation of Shanxi Province, China (Grant No.201601D102004)and the Natural Science Foundation for Young Scientistsof Shanxi Province, China (Grant No. 201601D021022),the Natural Science Foundation of Datong city (Grant No.20150110).
References