1 Introduction

The study of the thermodynamic characteristics of de Sitter space-time has arouse extensive attention in the recent years.[1−17]At the stage of cosmological inflation in the early time,our universe behaves like a quasi-de Sitter space-time,in which the cosmological constant takes the form of vacuum energy.Moreover,if the dark energy is simply a cosmological constant,i.e.,a component with constant equation of state,our universe will evolve into a new stage of de Sitter space-time in this simplest scenario.Therefore,a better knowledge of de Sitter space-time(especially its classical and quantum characteristics)is very important to construct the general framework of cosmic evolution.In the previous works,the black hole horizon and the cosmological horizon are always treated as two independent thermodynamic systems,[4−7,13]from which the thermodynamic volume of de Sitter space-time as well as the corresponding thermodynamic quantities satisfying the first thermodynamics law were obtained.[3]It is commonly recognized that the entropy of de Sitter space-time is the sum of that for the two types of horizons,[7,14]however,such statement concerning the nature of de Sitter space-time entropy still needs to be checked with adequate physical explanation.

在2017—2018秋季学期“生物化学”课程的教学中，采用“大班授课、小班研讨”、“设计问题和案例”、“网络平台实现师生互动”、“改变成绩评价体系”等方面的教学改革，在提高大班型授课效率方面取得了一定成效，并对上课模式和效果进行了无记名问卷调查，向参与的学生（参与传统教学问卷为120人，参与教学改革后问卷为110人）提出了5个问题。

Considering the fact that all thermodynamic quantities related to the black hole horizon and the cosmological horizon in de Sitter space-time can be expressed as a function of mass M,electric charge Q,and cosmological constant Λ,it is natural to consider the dependency between the two types of thermodynamic quantities.More specifically,the discussion of the following two problems is very significant to study the stability and evolution of de Sitter space-time:Do the thermodynamic quantities follow the behavior of their counterparts in AdS black holes,especially when the black hole horizon is correlated with that of the cosmological horizon?What is the specific relation between the entropy of de Sitter space-time and that of the two horizons(the black hole horizon and the cosmological horizon)? The above two problems also provide the main motivation of this paper.

库存管理，是对制造业或服务业生产、经营全过程的各种物品，产成品以及其他资源进行管理和控制，使其储备保持在经济合理的水平上。库存管理的内容包含仓库管理和库存控制。仓库管理的内容是库存物料的科学保管，以减少损耗，方便存取；库存控制则是要求控制合理的库存水平，即用最少的投资和最少的库存管理费用，维持合理的库存，以满足使用部门的需求和减少缺货损失，合理的库存是在满足需求的情况下尽量减少库存量，以降低资金的占用、减少存放保管等费用。

On the other hand,to the present,massive gravity has already proven its potential to alleviate the naturalness problem of the cosmological constant[18−20]even in the limit as the graviton mass approaches zero,which can be clearly seen both in the framework of linear gravity theories[21−22]and nonlinear massive gravity theories.[23−24]The ghostlike instability in the latter,which is well known as the Boulware-Deser ghost,has also been extensively discussed in many previous works.[25−26]Moreover,different from the traditional way of simply adding a mass term to the GR action,the new idea of constructing massive gravity theories was fi rstly studied in the context of a charged BTZ black hole,[27]and further well developed in the presence of Maxwell and Born-Infeld electrodynamics in asymptotically AdS space-times.[28]Details of the derived thermodynamic quantities(the corrected entropy,etc.)as well as the corresponding phase transitions may be found in Refs.[29–33].In this paper we expect to provide complementary results on the thermodynamics of dS black hole in massive gravity.

甲状腺功能亢进症通常多发于中年女性,是甲状腺激素产生过多引起的消化、神经等系统代谢亢奋和兴奋性提高的综合征,简称甲亢[1]。造成甲亢的原因复杂,如今在临床上没有明确的说明[2]。临床上常使用抗甲状腺药物和131 I进行治疗,效果明显,但是会产生一些不良反应[3]。研究表明,使用中西医结合对甲亢患者进行治疗效果显著,安全性高[4]。为了研究中西医结合治疗对于甲状腺功能亢进症患者的临床疗效及安全观察,本文选取甲状腺功能亢进症患者64例进行研究,随机分为对照组和观察组。比较两组患者的临床疗效、甲状腺功能指标、不良反应情况。

Following this direction,in our analysis we obtain the effective thermodynamic quantities of de Sitter black holes in massive gravity(DSBHMG),based on the correlation between the black hole horizon and the cosmological horizon. Our results show that the entropy of this type of space-time takes the same form as that in Reissner-Nordstrom de Sitter space-time,which lays a solid foundation for deeply understanding the universal thermodynamic characteristics of de Sitter space-time in the future.This paper is organized as follows.In Sec.2,we briefly introduce the thermodynamic quantities of the horizons of black holes and the Universe in DSBHMG,and furthermore obtain the electric charge Q when the two horizons show the same radioactive temperature.In Sec.3,taking the correlation between the two horizons into consideration,we will present the equivalent thermodynamic quantities of DSBHMG satisfying the first thermodynamic law,and perform a quantitative analysis of the corresponding effective temperature and pressure.Finally,the main conclusions are summarized in Sec.4.Throughout the paper we use the units G=~=kB=c=1.

2 Thermodynamics of Black Holes in Massive Gravity

In the framework of(3+1)-dimensional massive gravity with a Maxwell field(denoting Fµν as the Maxwell field-strength tensor),the corresponding action always expresses as[34−37]

理论层面，现有研究对国际舆论态势的基本判断仍是“英语霸权”“西强东弱”。国际传播的信息流亦呈现从西向东、由北到南、从发达国家流向发展中国家的状态。[11]在“一带一路”背景下，在国际舆论的话语实践中，中国需要调整以往在国际传播中的边缘心态，增强主动传播的意识，从被动的对媒介帝国主义的抵抗，转向积极地具有传播主体意识的议题建构，[12]从而形成“融通中外的话语体系”。培养合作传播思维，通过媒介合作共同打造话语空间，与周边国家和发展中国家构建话语共同体。未来的国际传播格局同样需要合作、融合、共赢的思维。相应地，未来的研究应关注如何通过媒介合作提高我国在国际传播中的主体地位问题。

where Λ is the cosmological constant,g and R are respectively the determinant of the metric and the Ricci scalar.The last term in the right hand side denotes the massive potential associate with graviton mass m,whereµi are symmetric polynomials of the eigenvalues representing the contribution of the matrixwith fixed symmetric tensor fµν.Generated form the above action,the space-time metric of static black holes(denoting hijas Einstein space with constant curvature)can be written as

When the temperature of the black hole horizon is equal to that of the cosmological horizon,the effective temperature of the space-time should be

Here k=+1,0,−1 respectively correspond to the sphere,Ricci flat,and hyperbolic symmetric cases.Note that the positions of black hole horizon r+and cosmic horizon rc are determined when f(r+,C)=0.

In Fig.1 we display the behavior of the metric function f(r),where the parameters are chosen as Λ=1,m0=30,m=2.12,c1=2,c2=3.18,q=1.7,while k is fixed at 1,0,and−1.It is obvious that there are two intersection points between f(r)and the axis of r,which respectively correspond to the positions of black hole horizon r+and cosmological horizon rc.Thus,the mass m0can be expressed in terms of r+,cas

where x=r+/rc.The temperature of the black hole horizons and cosmic horizon can be written as[40]

where

According to the Hamiltonian approach,we have the mass M and electric charge Q as

Then substituting Eq.(14)into Eq.(20),we get

茅台酒厂的科研机构经历了从无到有、从小到大的过程。从因陋就简到拥有多种现代化科研设备、科研手段和相当规模的科研机构，茅台在许多项目研究上都取得重大突破，研究硕果累累。

where ν2is the area of a unit volume of constant(t,r)space(which equals to 4π for k=0).

Fig.1 The metric function f(r)varying with r.

It is apparent that the thermodynamic quantities corresponding to the two horizons satisfy the first law of thermodynamics

where

①关于“法律规定的机关”。从现行法律体系和有关法理上看，水资源保护公益诉讼中 “法律规定的机关”应当仅限定为各级水行政主管部门和检察机关。检察机关作为法律监督机关，当然具有提起公益诉讼的主体资格。水行政主管部门之所以应当具有水资源保护公益诉讼主体资格，理由：一是水行政主管部门具有保护水事公共利益的法定职责，基于保护水事公共利益考虑理应有权提起公益诉讼；二是水行政主管部门具有启动公益诉讼程序的便利性和及时性，在发现破坏水事公共利益案件、及时启动公益诉讼程序方面，具有天然优势；三是水行政主管部门具有设备、专业知识以及能力方面的优势，更易于通过诉讼维护水事公共利益。

In Figs.2 and 3,we illustrate an example of the Pe ff-x diagram with different value of relevant parameters,from which one could clearly see the effect of these parameters on the effective pressure of RN-dSQ space-time.Following the same procedure by inserting Eq.(28)into Eq.(17),we can also obtain the S(x)-x and f(x)-x diagrams with rc=1,which are explicitly shown in Fig.4.Similarly,in Figs.5 and 6,we show the evolution of the Te ff-x diagram with different value of relevant parameters,from which one could perceive the effect of these parameters on the effective temperature of RN-dSQ space-time.As can be seen from Eq.(29),the special term m2c1and m2c2included in the expression of Te ffand Pe ff,could have significant effects in the evolution of the globally effective temperature and pressure.More specifically,it is shown in Figs.3 and 6 that,the space-time considered in our analysis will recover to the well-known RNdS spacetime when m2c1and m2c2approach zero.Such conclusion is well consistent with that obtained in the previous literature.[33]

When the temperature of the black hole horizon is equal to that of the cosmological horizon,the electric charge Q and the cosmological constant Λ are related as

As can be seen from Eqs.(5)and(12)the electric charge of the system satisfies the following expression

When taking T+=Tc,the combination of Eqs.(5),(6),and(13)will lead to the temperature T as

3 Effective Thermodynamic Quantities

where

where the thermodynamic volume is defined by[3,5−6,41]

It is obvious that there exit three real roots for the equation f(r)=0:the cosmological horizon(CEH)r=rc,the inner(Cauchy)horizon of black holes,and the outer horizon(BEH)r=r+of black holes.Moreover,the de Sitter space-time is characterized by Λ >0,while Λ <0 denotes the anti-de Sitter scenario.

Now we will consider the interaction between the black hole horizon and the cosmological horizon[42−43]

Here the undefined function f(x)represents the extra contribution from the correlations of the two horizons.We remark here that,when the black hole horizon and the cosmological horizon are treated as one thermodynamic system,the function f(x)could quantify the corresponding correction in the total entropy of the system.From Eq.(15),we can obtain the effective temperature Te ffand pressure Pe ff

Combining Eqs.(4),(16),and(17),one can obtain

Considering the connection between the black hole horizon and the cosmological horizon,we can derive the effective thermodynamic quantities and corresponding first law of black hole thermodynamics as

with the metric function expressed as[38−39]

and the entropy of the two horizons respectively express as

Turning to the contribution of the electrical charge q,it will also generate a chemical potential as

Then Eq.(25)will transform into

with the corresponding solution as

Considering the initial condition of f(0)=0,we can obtain

and inserting Eq.(25)into Eqs.(20)and(22)will lead to

Moreover,the effective chemical potential in the first law of black hole thermodynamics can be rewritten as

One could note that the effective chemical potential,ϕe ff,is not directly related to m2c1and m2c2,which was noted in the previous work discussing the entropy of RNdS black hole.[33]Based on the above equations,the Pe ff-x and Te ffx diagrams could be derived by taking different value of k,q,m,c1,and c2(when taking rc=1).

Fig.2 The Pe ff-x diagram when the parameter k is fixed at 1,0 and−1,respectively.The other parameters are fixed at m=2.12,c1=2,c2=3.18,q=1.7.

Fig.3 The Pe ff-x diagram varying with the parameters of m,c1,c2,and q,while the other two parameters are fixed at k=−1,rc=1.

Fig.4 The S(x)-x and f(x)-x diagrams with rc=1.

这款煮锅的手柄处设计有一个出水口，整个中空手柄可作为一个出水管，再配合一个过滤网，可将煮锅里的水从食物中过滤出来，很适合油炸食物将油与食物分离。不过，设计师可能忽略了中空手柄的清洁问题，里面日积月累的污垢如果不清洁会变得很不卫生。

模型3：由于爆轰产物的爆热难以估算，为此，模型3考虑用动量守恒的方法，建立第1阶段的速度v0计算模型。

Fig.6 The Te ff-x diagram varying with the parameters of m,c1,c2and q,while the other two parameters are fixed at k=−1,rc=1.

Table 1 Summary of the highest effective temperature and the corresponding xcfor different curves in Fig.5.The value of x0when the effective temperature reaches zero is also listed.

Parametric xc T2e ff x 0 k=1 0.3374 1.7554 0.2173 k=0 0.3468 1.5799 0.2243 k=−1 0.3571 1.4121 0.2321

Table 2 Summary of the highest effective temperature and the corresponding xcfor different curves in Fig.6.The value of x0when the effective temperature reaches zero is also listed.

Parametric xc T2e ff x0 q=1.0 0.222 69 2.922 26 0.134 98 k=−1 q=1.7 0.357 10 1.412 10 0.232 10 q=2.7 0.531 08 0.568 70 0.380 43 m=1.6 0.465 46 0.439 60 0.321 08 k=−1 m=2.12 0.357 10 1.412 10 0.223 10 m=3.12 0.249 46 5.611 53 0.153 24 c1=1.0 0.363 73 1.231 66 0.238 98 k=−1 c1=2.0 0.357 10 1.412 10 0.232 10 c1=3.0 0.351 14 1.595 04 0.225 99 c2=2.18 0.417 92 0.757 68 0.279 26 k=−1 c2=3.18 0.357 10 1.412 10 0.232 10 c2=4.18 0.316 77 2.224 55 0.202 18

4 Conclusion and Discussion

In this paper,by taking de Sitter space-time as a thermodynamic system,we study the effective thermodynamic quantities of de Sitter black holes in massive gravity,and furthermore obtain the effective thermodynamic quantities of the space-time.Here we summarize our main conclusions in more detail:

(i)In the previous analysis without considering the correlation between the black hole horizon and the cosmological horizon,i.e.,the two horizons are always treated as independent thermodynamic systems with different temperature,the space-time does not satisfy the requirement of thermodynamic stability.In this paper,we find that the establishment of the correlation between the two horizons will generate the common effective temperature Te ff,which may represent the most typical thermodynamic feature of RN-dSQ space-time.

(ii)As can be clearly seen from the S(x)-x and Te ff-x diagrams,RN-dSQ space-time in unstable under the condition of x>xcand x0.This result indicates that there exist only the RN-dSQ black holes satisfying the condition of x0c,which lays a solid theoretical foundation for the search of black holes in the Universe.

(iii)We find that the interaction term f(x)in the entropy of RN-dSQ space-time takes the same form of that in RN-dS space-time.Considering that the entropy in the two types of space-time is the function of the position of the horizon,which has no relation with other parameters including the electric charge(Q)and the constant(Λ),the entropy in the two types of space-time should take the same form.This finding may contribute to the deep understanding the universal thermodynamic characteristics of de Sitter space-time in the future.

[12]Still,Pakistani economists disagree as to whether their country can fully take advantage of the opportunity.Some note that it is unclear whether the agreement will help Pakistan overcome a 50 percent trade imbalance with China.

Acknowledgments

云锡文山锌铟冶炼技改项目，是云南省“四个一百”重点建设项目。中国恩菲按照国内领先、世界一流的标准开展项目设计，针对低锌、高铟、高铁的原料特性，联合业主和相关单位，开展了坚持不懈的科技攻关，成功研发出赤铁矿除铁新工艺，突破了技术难关、打破了国外技术垄断，填补了国内空白，使赤铁矿除铁新工艺在国内首次得到大规模工业化应用，也成为锌冶炼领域世界上第三个采用此工艺的项目。项目的投产，标志着我国锌冶炼工艺新时代的到来。

The authors declare that there is no conflict of interest regarding the publication of this paper.

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