research article the critical phenomena and thermodynamics...

Research ArticleThe Critical Phenomena and Thermodynamics ofthe Reissner-Nordstrom-de Sitter Black Hole

Ren Zhao12 Mengsen Ma12 Huihua Zhao12 and Lichun Zhang12

1 Institute of Theoretical Physics Shanxi Datong University Datong 037009 China2Department of Physics Shanxi Datong University Datong 037009 China

Correspondence should be addressed to Ren Zhao zhao2969sinacom and Mengsen Ma mengsenmagmailcom

Received 25 October 2013 Revised 26 January 2014 Accepted 5 February 2014 Published 10 March 2014

Academic Editor Deyou Chen

Copyright copy 2014 Ren Zhao et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

It is wellknown that there are two horizons for the Reissner-Nordstrom-de Sitter spacetime namely the black hole horizon and thecosmological one Both horizons can usually seem to be two independent thermodynamic systems however the thermodynamicquantities on both horizons satisfy the laws of black hole thermodynamics and are not independent In this paper by consideringthe relations between the two horizons we give the effective thermodynamic quantities in Reissner-Nordstrom-de Sitter spacetimeThe thermodynamic properties of these effective quantities are analyzed moreover the critical temperature critical pressure andcritical volume are obtained We also discussed the thermodynamic stability of Reissner-Nordstrom-de Sitter spacetime

1 Introduction

Black hole physics especially the black hole thermodynamicsrefer directly to the theories of gravity statistical physicsparticle physics field theory and so forth This makes thefield concerned by many physicists [1ndash6] Although the com-plete statistical description of black hole thermodynamicsis still unclear the research on the properties of black holethermodynamics is prevalent such as Hawking-Page phasetransition [7] and critical phenomenaMore interestingly theresearch on the charged and nonrotating RN-AdS black holeshows that there exists a similar phase transition to the vander Waals-Maxwell vapor-liquid phase transition [8 9]

Motivated by the AdSCFT correspondence [10] wherethe transitions have been related with the holographic super-conductivity [11 12] the subject of the phase transitions ofblack holes in asymptotically anti-de Sitter (AdS) spacetimehas received considerable attention [13ndash17] The underlyingmicroscopic statistical interaction of the black holes is alsoexpected to be understood via the study of the gauge theoryliving on the boundary in the gaugegravity duality

Recently by considering the cosmological constant corre-spond to pressure in general thermodynamic system namely

119875 = minus1

8120587Λ =

3

8120587

1

1198972 (1)

the thermodynamic volumes in AdS and dS spacetime areobtained [18ndash24] The studies on phase transition of blackholes have aroused great interest [25ndash31] Connecting thethermodynamic quantities of AdS black holes to (119875 sim 119881) inthe ordinary thermodynamic system the critical behaviors ofblack holes can be analyzed and the phase diagram like vander Waals vapor-liquid system can be obtained This helpsto further understand the black hole entropy temperatureheat capacities and so forth It also has a very importantsignificance in completing the geometric theory of black holethermodynamics

As is well known there are black hole horizon and cosmo-logical horizon in the appropriate range of parameters for deSitter spacetime Both horizons have thermal radiation butwith different temperatures The thermodynamic quantitieson both horizons satisfy the first law of thermodynamics

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2014 Article ID 124854 6 pageshttpdxdoiorg1011552014124854

2 Advances in High Energy Physics

and the corresponding entropy fulfills the area formula [2332 33] In recent years the research on the thermodynamicproperties of de Sitter spacetime has drawn a lot of attention[23 32ndash36] In the inflation epoch of early universe theuniverse is a quasi-de Sitter spacetime The cosmologicalconstant introduced in de Sitter space may come fromthe vacuum energy which is also a kind of energy If thecosmological constant is the dark energy the universe willevolve to a new de Sitter phase To depict the whole history ofevolution of the universe we should have some knowledgeon the classical and quantum properties of de Sitter space[23 33 37 38]

Firstly we expect the thermodynamic entropy to satisfythe Nernst theorem [34 35 39] At present a satisfactoryexplanation to the problem in which the thermodynamicentropy of the horizon of the extreme de Sitter spacetime doesnot fulfill the Nernst theorem is still lacking Secondly whenconsidering the correlation between the black hole horizonand the cosmological horizon whether the thermodynamicquantities in de Sitter spacetime still have the phase transitionand critical behavior like in AdS black holes Thus it isworthy of our deep investigation and reflection to establisha consistent thermodynamics in de Sitter spacetime

Because the thermodynamic quantities on the black holehorizon and the cosmological one in de Sitter spacetime arethe functions of mass119872 electric charge119876 and cosmologicalconstant Λ The quantities are not independent of eachother Considering the relation between the thermodynamicquantities on the two horizons is very important for studyingthe thermodynamic properties of de Sitter spacetime Basedon the relation we give the effective temperature and pressureof Reissner-Nordstrom-de Sitter(R-NdS) spacetime and ana-lyze the critical behavior of the equivalent thermodynamicquantities It is shown that when considering the relationbetween the two horizons in RN-dS spacetime there is thesimilar phase transition like the ones in van derWaals liquid-gas system and charged AdS black holes

The paper is arranged as follows In Section 2 we intro-duce the Reissner-Nordstrom-de Sitter(R-NdS) spacetimeand give the two horizons and corresponding thermody-namic quantities In Section 3 by considering the relationsbetween the two horizons we obtain the effective temperatureand the equivalent pressure In Section 4 the critical phe-nomena of effective thermodynamic quantities are discussedFinally we discuss and summarize our results in Section 5 (weuse the units 119866

119899+1= ℎ = 119896

119861= 119888 = 1)

2 RN-dS Spacetime

The line element of the R-N SdS black holes is given by [32]

1198891199042= minus119891 (119903) 119889119905

2+ 119891minus11198891199032+ 1199032119889Ω2 (2)

where

119891 (119903) = 1 minus2119872

119903+1198762

1199032minusΛ

31199032 (3)

The above geometry possesses three horizons the black holeCauchy horizon located at 119903 = 119903

minus the black hole event

horizon (BEH) located at 119903 = 119903+ and the cosmological event

horizon (CEH) located at 119903 = 119903119888 where 119903

119888gt 119903+gt 119903minus the only

real positive zeroes of 119891(119903) = 0The equations 119891(119903

+) = 0 and 119891(119903

119888) = 0 are rearranged to

1198762= 119903+119903119888(1 minus

1199032

119888+ 119903119888119903++ 1199032

+

3Λ)

2119872 = (119903119888+ 119903+) (1 minus

1199032

119888+ 1199032

+

3Λ)

(4)

The surface gravity on the black hole horizon and thecosmological horizon is respectively

120581+=1

2

119889119891 (119903)

119889119903

10038161003816100381610038161003816100381610038161003816119903=119903+

=1199032

+minus 1198762minus 1199034

+Λ

21199033+

=(119903+minus 119903119888)

21199032+

(1 minus

(1199032

119888+ 2119903+119903119888+ 31199032

+) (119903119888119903+minus 1198762)

119903119888119903+(1199032119888+ 119903+119903119888+ 1199032+)

)

120581119888=1

2

119889119891 (119903)

119889119903

10038161003816100381610038161003816100381610038161003816119903=119903119888

=1199032

119888minus 1198762minus 1199034

119888Λ

21199033119888

=(119903119888minus 119903+)

21199032119888

(1 minus

(1199032

++ 2119903+119903119888+ 31199032

119888) (119903119888119903+minus 1198762)

119903119888119903+(1199032119888+ 119903+119903119888+ 1199032+)

)

(5)

The thermodynamic quantities on the two horizons satisfythe first law of thermodynamics [23 33 40 41]

120575119872 =120581+

2120587120575119878++ Φ+120575119876 + 119881

+120575119875

120575119872 =120581119888

2120587120575119878119888+ Φ119888120575119876 + 119881

119888120575119875

(6)

where 119878+= 1205871199032

+ 119878119888= 1205871199032

119888 Φ+= 119876119903

+ Φ119888= minus(119876119903

119888) 119881+=

(41205873)1199033

+ 119881119888= (41205873)119903

3

119888 119875 = minus(Λ8120587)

3 Thermodynamic Quantity ofRN-dS Spacetime

In Section 2 we have obtained thermodynamic quantitieswithout considering the relationship between the blackhole horizon and the cosmological horizon Because thereare three variables 119872 119876 and Λ in the spacetime thethermodynamic quantities corresponding to the black holehorizon and the cosmological horizon are functional withrespect to 119872 119876 and Λ The thermodynamic quantitiescorresponding to the black hole horizon are related to theones corresponding to the cosmological horizon When thethermodynamic property of charged de Sitter spacetime isstudied we must consider the relationship with the twohorizons Recently by studyingHawking radiation of de Sitter

Advances in High Energy Physics 3

spacetime [42 43] obtained that the outgoing rate of thecharged de Sitter spacetime which radiates particles withenergy 120596 is

Γ = 119890Δ119878++Δ119878119888 (7)

where Δ119878+and Δ119878

119888are Bekenstein-Hawking entropy dif-

ference corresponding to the black hole horizon and thecosmological horizon after the charged de Sitter spacetimeradiates particles with energy 120596 Therefore the thermody-namic entropy of the charged de Sitter spacetime is the sumofthe black hole horizon entropy and the cosmological horizonentropy

119878 = 119878++ 119878119888 (8)

Substituting (6) into (8) one can obtain

119889119878 = 2120587(1

120581+

+1

120581119888

)119889119872

minus 2120587(120593+

120581+

+120593119888

120581119888

)119889119876 +120587

3(1199033

+

120581+

+1199033

119888

120581119888

)119889Λ

(9)

For simplicity we consider the case with constant In this casethe equation above turns into

119889119878 = 2120587(1

120581+

+1

120581119888

)119889119872 +120587

3(1199033

+

120581+

+1199033

119888

120581119888

)119889Λ (10)

From (4) and (5) we derive

119889119903119888=119889119872

119903119888120581119888

+

(1199033

1198883)

2119903119888120581119888

119889Λ 119889119903+=119889119872

119903+120581+

+

(1199033

+3)

2119903+120581+

119889Λ

(11)

Recently the thermodynamic volume of RN-dS spacetime[23 34] is given as

119881 =4120587

3(1199033

119888minus 1199033

+) (12)

Substituting (11) into (12) one gets

119889119881 = 4120587(119903119888

120581119888

minus119903+

120581+

)119889119872 +2120587

3(1199034

119888

120581119888

minus1199034

+

120581+

)119889Λ (13)

Substituting (13) into (10) one can derive the thermodynamicequation of thermodynamic quantities in de Sitter spacetime[34]

119889119872 = 119879eff119889119878 minus 119875eff119889119881 (14)

where the effective temperature is

119879eff =(1199094+ 1199093minus 21199092+ 119909 + 1)

4120587119903119888119909 (119909 + 1) (119909

2 + 119909 + 1)

minus1198762

412058711990331198881199093 (119909 + 1) (119909

2 + 119909 + 1)

times (1 + 119909 + 1199092minus 21199093+ 1199094+ 1199095+ 1199096)

(15)

The effective pressure is

119875eff =(1 minus 119909) (1 + 3119909 + 3119909

2+ 31199093+ 1199094)

81205871199032119888119909 (1 + 119909) (1 + 119909 + 119909

2)2

minus

1198762(1 minus 119909) (1 + 2119909 + 3119909

2minus 31199095minus 21199096minus 1199097)

812058711990341198881199093 (1 + 119909) (1 minus 119909

3) (1 + 119909 + 1199092)

(16)

where 119909 = 119903+119903119888and 0 lt 119909 lt 1

From (15) and (16) when119876 = 0 the effective temperatureand pressure are both greater than zeroThis fulfills the stablecondition of thermodynamic equilibrium If considering theblack hole horizon and the cosmological one as independentof each other because of the different radiant temperatureson the two horizons the spacetime is instable

Another problem of considering the black hole horizonand the cosmological one as independent of each other is thatwhen the two horizons coincide namely

1199032

+119888=1 plusmn radic1 minus 41198762Λ

2Λ= 1199032

0 (17)

from (4) the surface gravity 120581+119888= 0 thus the temperature

on the black hole horizon and the temperature on thecosmological horizon are both zero However both horizonshave nonzero area which means that the entropy for the twohorizons should not be zero This conclusion is inconsistentwith Nernst theorem In the extreme case 1199032

+119888= 1199032

0 the

effective temperature from (15) is

119879eff =1

121205871199030

(1 minus51198762

21199032

0

) (18)


119875eff = 0 (19)

In this case the volume-thermodynamic system becomesarea-thermodynamic one According to (19) the pressure ofthermodynamic membrane is zero However from (18) thetemperature of thermodynamic membrane is nonzero Thiscan partly solve the problem that extreme de Sitter black holesdo not satisfy the Nernst theorem when 119876 = 0 (15) and (16)return to the known result [34]

4 Critical Behaviour

To compare with the van der Waals equation we set 119875eff rarr119875 V rarr V and discuss the phase transition and the criticalphenomena when 119903

119888is invariantThe van derWaals equation

is

(119875 +119886

V2) (V minus ) = 119896119879 (20)

Here V = 119881119873 is the specific volume of the fluid 119875 isits pressure 119879 is its temperature and 119896 is the Boltzmannconstant

Substituting (15) into (16) we obtain

119875eff = 119879eff1198614

21199031198881198612

+11986121198613minus 11986111198614

81205871199032119888119909 (1 + 119909) 119861

2

(21)


where

1198611=1 + 119909 minus 2119909

2+ 1199093+ 1199094

1 + 119909 + 1199092

1198612=1 + 119909 + 119909

2minus 21199093+ 1199094+ 1199095+ 1199096

1 + 119909 + 1199092

1198613=1 + 2119909 minus 2119909

4minus 1199095

(1 + 119909 + 1199092)2

1198614=1 + 2119909 + 3119909

2minus 31199095minus 21199096minus 1199097

(1 + 119909 + 1199092)2

(22)

According to (21) combing dimensional analysis [22]with (12) we conclude that we should identify the specificvolume V with

V = 119903119888(1 minus 119909) (23)

From the two equations

120597119875eff120597V

= 01205972119875eff120597V2

= 0 (24)

we first calculate the position of critical pointsThen one canderive

(120597119875eff120597V)

119879eff

= minus1 times (81205871199031198885(1 + 119909) (1 + 119909 + 119909

2)3

times (1 + 119909 + 1199092minus 21199093+ 1199094+ 1199095+ 1199096) )

minus1

times [11990311988821199092(minus6 minus 21119909 minus 38119909

2minus 45119909

3minus 30119909

4minus 10119909

5

+ 61199096+ 71199097+ 41199098+ 1199099)

times 1198762119909 (15 + 54119909 + 98119909

2+ 104119909

3+ 59119909

4+ 18119909

5

minus 91199096minus 12119909

7minus 10119909

8minus 41199099minus 11990910)] = 0

(1205972119875eff120597V2

)

119879eff

= 1 times (41205871199031198886(1 + 119909) (1 + 119909 + 119909

2)4

times (1 + 119909 + 1199092minus 21199093+ 1199094+ 1199095+ 1199096)2

)

minus1

times [minus1199031198882119909 (6 + 36119909 + 115119909

2+ 286119909

3+ 486119909

4

+ 5301199095+ 203119909

6minus 357119909

7minus 783119909

8

minus 8541199099minus 666119909

10minus 375119909

11minus 127119909

12

+311990913+ 32119909

14+ 18119909

15+ 611990916+ 11990917)

+ 1198762(30 + 165119909 + 456119909

2+ 837119909

3+ 1050119909

4

+ 9031199095+ 255119909

6minus 720119909

7minus 1707119909

8

minus 21131199099minus 1743119909

10minus 969119909

11minus 314119909

12

minus 911990913+ 63119909

14+ 44119909

15+ 21119909

16+ 611990917

+ 11990918)] = 0

(25)

When 119903119888= 1 from (15) (16) and (25) one can obtain the

position of critical point

119909119888= 0871992 (26)

The critical electric charge specific volume temperature andthe critical pressure are respectively

119876119888= 060395 V119888 = 0128008

119879119888= 000436559 119901

119888= 00000145468

(27)

In Figure 1 we give the figure of 119875eff with the change of Vat the constant effective temperature near the critical point

To reflect the influence of 119903119888on the spacetime we set 119903

119888=

5 from which we derived the position of the critical point

119909119888= 0871992 (28)


119876119888= 301975 V119888 = 0640038

119879119888= 0000873119 119901

119888= 58187 times 10

minus7

(29)

The diagram of the effective 119875eff with the change of V isdepicted in Figure 2

For the van der Waals fluid and the RN-AdS black holethe relation 119875

119888V119888119879119888is a universal number and is independent

of the charge 119876 For the RN-dS black hole according to theeffective thermodynamic quantities numerical calculationshows that 119875

119888V119888119879119888still has a 119876-independent universal value

Certainly the universal value is no more than 38 butsim00004265

5 Discussion and Conclusions

From above we can find out that the value of 119903119888does

not influence the critical point 119909 namely for the RN-dSspacetime the position of critical point is irrelevant to thevalue of the cosmological horizon This indicates that 119909 =

119903119888119903+is fixed however the critical effective temperature

critical volume and critical pressure are dependent on thevalue of 119903

119888 The critical effective temperature and pressure for

the RN-dS system will decrease as the values of 119903119888increase

while the critical electric charge and the critical volume willincrease as the values of 119903

119888increase

By Figures 1 and 2 the 119875eff minus V curve at constanttemperature of RN-dS spacetime is different from the onesof van der Waals equation and charged AdS black holes


02 04 06 08 100000

0001

0002

0003

0004

0005

minus0001

minus0002

Peff

Figure 1 119875eff minus V diagram of RN-dS black holes The temperatureof isotherms decreases from top to bottom and corresponds to 119879119888 +0004 119879119888 + 0002 119879119888 119879119888 minus 0002

15

1

05

0

minus05

times10minus6

02 04 06 08 10

Peff

2

25

3


(1) The first difference lies at the critical pressure 119875effof RN-dS spacetime which increases as the volumeincreases at the constant temperature when 119879eff gt 119879

119888and themaximal value turns up at 119909 rarr 0 namely deSitter spacetime

(2) The first difference is that the critical pressure 119875effof RN-dS spacetime may be negative at the constanttemperature when 119879eff lt 119879

119888 which means the systemis not stable From (15) and (16) we can obtain theparameters119909 and119876when the system is in the unstablestate

(3) On the 119875eff minus V curves with 119879eff gt 119879119888 the system lies

in a phase on the 119875eff minus V curves with 119879eff lt 119879119888 the

same pressure will correspond to two different valuesof 119909 namely two-phase coexistence region Using theequal area criterion we can find out the proportion ofthe two phases for the RN-dS system

From the above discussion in RN-dS spacetime whenconsidering the relation between the two horizons there isthe similar phase transition like the one in van der Waalsequation and charged AdS black holes The reason is stillunclear This deserves further study If the cosmologicalconstant is just the dark energy the universe will evolve intoa new de Sitter phase According to Figures 1 and 2 the RN-dS system can evolve into de Sitter phase along isothermalcurve However along the different isothermal curves whichprocesses will be the evolution of the universe to a new deSitter phase needs further consideration according to theobservations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by NSFC under Grant nos 11175109and 11075098 and by the Doctoral Sustentation Foundationof Shanxi Datong University (2011-B-03)

References

[1] J D Bekenstein ldquoBlack holes and the second lawrdquo Lettere AlNuovo Cimento vol 4 no 15 pp 737ndash740 1972

[2] J D Bekenstein ldquoGeneralized second law of thermodynamicsin black-hole physicsrdquoPhysical ReviewD vol 9 no 12 pp 3292ndash3300 1974

[3] J D Bekenstein ldquoExtraction of energy and charge from a blackholerdquo Physical Review D vol 7 no 4 pp 949ndash953 1973

[4] J M Bardeen B Carter and S W Hawking ldquoThe four lawsof black hole mechanicsrdquo Communications in MathematicalPhysics vol 31 pp 161ndash170 1973

[5] S W Hawking ldquoBlack hole explosionsrdquo Nature vol 248 no5443 pp 30ndash31 1974

[6] S W Hawking ldquoParticle creation by black holesrdquo Communica-tions in Mathematical Physics vol 43 no 3 pp 199ndash220 1975

[7] S W Hawking and D N Page ldquoThermodynamics of blackholes in anti-de Sitter spacerdquo Communications in MathematicalPhysics vol 87 no 4 pp 577ndash588 1983

[8] A Chamblin R Emparan C V Johnson and R C MyersldquoCharged AdS black holes and catastrophic holographyrdquo Physi-cal Review D vol 60 no 6 Article ID 064018 1999

[9] A Chamblin R Emparan C V Johnson and R C MyersldquoHolography thermodynamics and fluctuations of chargedAdS black holesrdquo Physical Review D vol 60 no 10 Article ID104026 1999

[10] O Aharony S S Gubser J Maldacena H Ooguri and YOz ldquoLarge 119873 field theories string theory and gravityrdquo PhysicsReport vol 323 no 3-4 pp 183ndash386 2000


[11] S S Gubser ldquoBreaking an Abelian gauge symmetry near ablack hole horizonrdquo Physical Review D vol 78 no 6 ArticleID 065034 2008

[12] S A Hartnoll C P Herzog and G T Horowitz ldquoBuildingan AdSCFT superconductorrdquo Physical Review Letters vol 101Article ID 031601 2008

[13] A Sahay T Sarkar and G Sengupta ldquoThermodynamic geom-etry and phase transitions in Kerr-Newman-AdS black holesrdquoJournal of High Energy Physics vol 2010 no 4 article 118 2010

[14] A Sahay T Sarkar and G Sengupta ldquoOn the thermodynamicgeometry and critical phenomena of AdS black holesrdquo Journalof High Energy Physics vol 2010 no 7 article 082 2010

[15] A Sahay T Sarkar and G Sengupta ldquoOn the phase structureand thermodynamic geometry of R-charged black holesrdquo Jour-nal of High Energy Physics vol 2010 no 11 article 125 2010

[16] R Banerjee S Ghosh and D Roychowdhury ldquoNew type ofphase transition in Reissner Nordstrom-AdS black hole and itsthermodynamic geometryrdquo Physics Letters B vol 696 no 1-2pp 156ndash162 2011

[17] D Kastor S Ray and J Traschen ldquoEnthalpy and the mechanicsof AdS black holesrdquo Classical and Quantum Gravity vol 26 no19 Article ID 195011 2009

[18] P Brian Dolan ldquoThe cosmological constant and the black holeequation of staterdquo Classical and Quantum Gravity vol 28Article ID 125020 2011

[19] S Gunasekaran D Kubiznak and R BMann ldquoExtended phasespace thermodynamics for charged and rotating black holesand Born-Infeld vacuum polarizationrdquo Journal of High EnergyPhysics vol 2012 article 110 2012

[20] B P Dolan ldquoCompressibility of rotating black holesrdquo PhysicalReview D vol 84 Article ID 127503 2011

[21] M Cvetic G W Gibbons D Kubiznak and C N Pope ldquoBlackhole enthalpy and an entropy inequality for the thermodynamicvolumerdquo Physical Review D vol 84 Article ID 024037 2011

[22] D Kubiznak and R B Mann ldquoP-V criticality of charged AdSblack holesrdquo Journal of High Energy Physics vol 1207 no 332012

[23] B P Dolan D Kastor D Kubiznak R BMann and J TraschenldquoThermodynamic volumes and isoperimetric inequalities for deSitter Black Holesrdquo Physical Review D vol 87 Article ID 1040172013

[24] B P Dolan ldquoWhere is the PdV in the first law of blackhole thermodynamicsrdquo in Open Questions in Cosmology JGonzalo Olmo Ed InTech 2012 httpwwwintechopencombooksopen-questions-in-cosmologywhere-is-the-pdv-term-in-the-first-law-of-black-hole-thermodynamics-

[25] W Shao-Wen and L Yu-Xiao ldquoCritical phenomena and ther-modynamic geometry of charged Gauss-Bonnet AdS blackholesrdquo Physical Review D vol 87 Article ID 044014 2013

[26] R-G Cai L-M Cao R-Q Yang et al ldquoPV criticality in theextended phase space of Gauss-Bonnet black holes in AdSspacerdquo Journal of High Energy Physics vol 2013 pp 1ndash22 2013

[27] S H Hendi and M H Vahidinia ldquoExtended phase space ther-modynamics and P-V criticality of black holes with nonlinearsourcerdquo Physical Review D vol 88 Article ID 084045

[28] A Belhaj M Chabab H E Moumni and M B SedraldquoCritical behaviors of 3D black holes with a scalar hairrdquohttparxivorgabs13062518

[29] R Zhao H H Zhao M S Ma and L C Zhang ldquoOn thermo-dynamics of charged and rotating asymptotically AdS blackstringsrdquoTheEuropean Physical Journal C vol 73 no 12 pp 1ndash102013

[30] M Bagher J Poshteh B Mirza and Z Sherkatghanad ldquoOnthe phase transition critical behavior and critical exponents ofMyers-Perry black holesrdquo Physical Review D vol 88 Article ID024005 2013

[31] E Spallucci and A Smailagic ldquoMaxwellrsquos equal area law forcharged Anti-de Sitter black holesrdquo Physics Letters B vol 723no 4-5 pp 436ndash441 2013

[32] R-G Cai ldquoCardy-Verlinde formula and thermodynamics ofblack holes in de Sitter spacesrdquo Nuclear Physics B vol 628 no1-2 pp 375ndash386 2002

[33] Y Sekiwa ldquoThermodynamics of de Sitter black holes thermalcosmological constantrdquo Physical Review D vol 73 no 8 ArticleID 084009 2006

[34] MUrano andA Tomimatsu ldquoMechanical first law of black holespacetimes with a cosmological constant and its application tothe Schwarzschild-de Sitter spacetimerdquo Classical and QuantumGravity vol 26 no 10 Article ID 105010 2009

[35] L C Zhang H F Li and R Zhao ldquoThermodynamics of theReissner-Nordstrom-de Sitter black holerdquo Science China vol54 no 8 pp 1384ndash1387 2011

[36] Y S Myung ldquoThermodynamics of the Schwarzschild-de Sitterblack hole thermal stability of the Nariai black holerdquo PhysicalReview D vol 77 no 10 Article ID 104007 2008

[37] R-G Cai Physics vol 34 p 555 2005 (Chinese)[38] S Bhattacharya and A Lahiri ldquoMass function and particle

creation in Schwarzschild-de Sitter spacetimerdquo The EuropeanPhysical Journal C vol 73 no 12 pp 1ndash10 2013

[39] R-G Cai J-Y Ji and K-S Soh ldquoTopological dilaton blackholesrdquo Classical and Quantum Gravity vol 15 no 9 pp 2783ndash2793 1998

[40] G W Gibbons H Lu D N Page and C N Pope ldquoRotatingblack holes in higher dimensions with a cosmological constantrdquoPhysical Review Letters vol 93 no 17 Article ID 171102 2004

[41] G W Gibbons H Lu D N Page and C N Pope ldquoThe generalKerr-de Sitter metrics in all dimensionsrdquo Journal of Geometryand Physics vol 53 no 1 pp 49ndash73 2005

[42] R Zhao L-C Zhang and H-F Li ldquoHawking radiation ofa Reissner-Nordstrom-de Sitter black holerdquo General Relativityand Gravitation vol 42 no 4 pp 975ndash983 2010

[43] Z Ren Z Lichun L Huaifan and W Yueqin ldquoHawkingradiation of a high-dimensional rotating black holerdquo EuropeanPhysical Journal C vol 65 no 1 pp 289ndash293 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


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and the corresponding entropy fulfills the area formula [2332 33] In recent years the research on the thermodynamicproperties of de Sitter spacetime has drawn a lot of attention[23 32ndash36] In the inflation epoch of early universe theuniverse is a quasi-de Sitter spacetime The cosmologicalconstant introduced in de Sitter space may come fromthe vacuum energy which is also a kind of energy If thecosmological constant is the dark energy the universe willevolve to a new de Sitter phase To depict the whole history ofevolution of the universe we should have some knowledgeon the classical and quantum properties of de Sitter space[23 33 37 38]

Firstly we expect the thermodynamic entropy to satisfythe Nernst theorem [34 35 39] At present a satisfactoryexplanation to the problem in which the thermodynamicentropy of the horizon of the extreme de Sitter spacetime doesnot fulfill the Nernst theorem is still lacking Secondly whenconsidering the correlation between the black hole horizonand the cosmological horizon whether the thermodynamicquantities in de Sitter spacetime still have the phase transitionand critical behavior like in AdS black holes Thus it isworthy of our deep investigation and reflection to establisha consistent thermodynamics in de Sitter spacetime

Because the thermodynamic quantities on the black holehorizon and the cosmological one in de Sitter spacetime arethe functions of mass119872 electric charge119876 and cosmologicalconstant Λ The quantities are not independent of eachother Considering the relation between the thermodynamicquantities on the two horizons is very important for studyingthe thermodynamic properties of de Sitter spacetime Basedon the relation we give the effective temperature and pressureof Reissner-Nordstrom-de Sitter(R-NdS) spacetime and ana-lyze the critical behavior of the equivalent thermodynamicquantities It is shown that when considering the relationbetween the two horizons in RN-dS spacetime there is thesimilar phase transition like the ones in van derWaals liquid-gas system and charged AdS black holes

The paper is arranged as follows In Section 2 we intro-duce the Reissner-Nordstrom-de Sitter(R-NdS) spacetimeand give the two horizons and corresponding thermody-namic quantities In Section 3 by considering the relationsbetween the two horizons we obtain the effective temperatureand the equivalent pressure In Section 4 the critical phe-nomena of effective thermodynamic quantities are discussedFinally we discuss and summarize our results in Section 5 (weuse the units 119866

119899+1= ℎ = 119896

119861= 119888 = 1)

2 RN-dS Spacetime

The line element of the R-N SdS black holes is given by [32]

1198891199042= minus119891 (119903) 119889119905

2+ 119891minus11198891199032+ 1199032119889Ω2 (2)

where

119891 (119903) = 1 minus2119872

119903+1198762

1199032minusΛ

31199032 (3)

The above geometry possesses three horizons the black holeCauchy horizon located at 119903 = 119903

minus the black hole event

horizon (BEH) located at 119903 = 119903+ and the cosmological event

horizon (CEH) located at 119903 = 119903119888 where 119903

119888gt 119903+gt 119903minus the only

real positive zeroes of 119891(119903) = 0The equations 119891(119903

+) = 0 and 119891(119903

119888) = 0 are rearranged to

1198762= 119903+119903119888(1 minus

1199032

119888+ 119903119888119903++ 1199032

+

3Λ)

2119872 = (119903119888+ 119903+) (1 minus

1199032

119888+ 1199032

+

3Λ)

(4)

The surface gravity on the black hole horizon and thecosmological horizon is respectively

120581+=1

2

119889119891 (119903)

119889119903

10038161003816100381610038161003816100381610038161003816119903=119903+

=1199032

+minus 1198762minus 1199034

+Λ

21199033+

=(119903+minus 119903119888)

21199032+

(1 minus

(1199032

119888+ 2119903+119903119888+ 31199032

+) (119903119888119903+minus 1198762)

119903119888119903+(1199032119888+ 119903+119903119888+ 1199032+)

)

120581119888=1

2

119889119891 (119903)

119889119903

10038161003816100381610038161003816100381610038161003816119903=119903119888

=1199032

119888minus 1198762minus 1199034

119888Λ

21199033119888

=(119903119888minus 119903+)

21199032119888

(1 minus

(1199032

++ 2119903+119903119888+ 31199032

119888) (119903119888119903+minus 1198762)

119903119888119903+(1199032119888+ 119903+119903119888+ 1199032+)

)

(5)

The thermodynamic quantities on the two horizons satisfythe first law of thermodynamics [23 33 40 41]

120575119872 =120581+

2120587120575119878++ Φ+120575119876 + 119881

+120575119875

120575119872 =120581119888

2120587120575119878119888+ Φ119888120575119876 + 119881

119888120575119875

(6)

where 119878+= 1205871199032

+ 119878119888= 1205871199032

119888 Φ+= 119876119903

+ Φ119888= minus(119876119903

119888) 119881+=

(41205873)1199033

+ 119881119888= (41205873)119903

3

119888 119875 = minus(Λ8120587)

3 Thermodynamic Quantity ofRN-dS Spacetime

In Section 2 we have obtained thermodynamic quantitieswithout considering the relationship between the blackhole horizon and the cosmological horizon Because thereare three variables 119872 119876 and Λ in the spacetime thethermodynamic quantities corresponding to the black holehorizon and the cosmological horizon are functional withrespect to 119872 119876 and Λ The thermodynamic quantitiescorresponding to the black hole horizon are related to theones corresponding to the cosmological horizon When thethermodynamic property of charged de Sitter spacetime isstudied we must consider the relationship with the twohorizons Recently by studyingHawking radiation of de Sitter



Γ = 119890Δ119878++Δ119878119888 (7)




119878 = 119878++ 119878119888 (8)


119889119878 = 2120587(1

120581+

+1

120581119888

)119889119872

minus 2120587(120593+

120581+

+120593119888

120581119888

)119889119876 +120587

3(1199033

+

120581+

+1199033

119888

120581119888

)119889Λ

(9)


119889119878 = 2120587(1

120581+

+1

120581119888

)119889119872 +120587

3(1199033

+

120581+

+1199033

119888

120581119888

)119889Λ (10)


119889119903119888=119889119872

119903119888120581119888

+

(1199033

1198883)

2119903119888120581119888

119889Λ 119889119903+=119889119872

119903+120581+

+

(1199033

+3)

2119903+120581+

119889Λ

(11)


119881 =4120587

3(1199033

119888minus 1199033

+) (12)


119889119881 = 4120587(119903119888

120581119888

minus119903+

120581+

)119889119872 +2120587

3(1199034

119888

120581119888

minus1199034

+

120581+

)119889Λ (13)


119889119872 = 119879eff119889119878 minus 119875eff119889119881 (14)


119879eff =(1199094+ 1199093minus 21199092+ 119909 + 1)

4120587119903119888119909 (119909 + 1) (119909

2 + 119909 + 1)

minus1198762

412058711990331198881199093 (119909 + 1) (119909

2 + 119909 + 1)

times (1 + 119909 + 1199092minus 21199093+ 1199094+ 1199095+ 1199096)

(15)


119875eff =(1 minus 119909) (1 + 3119909 + 3119909

2+ 31199093+ 1199094)

81205871199032119888119909 (1 + 119909) (1 + 119909 + 119909

2)2

minus

1198762(1 minus 119909) (1 + 2119909 + 3119909


812058711990341198881199093 (1 + 119909) (1 minus 119909

3) (1 + 119909 + 1199092)

(16)

where 119909 = 119903+119903119888and 0 lt 119909 lt 1



1199032


2Λ= 1199032

0 (17)



+119888= 1199032

0 the


119879eff =1

121205871199030

(1 minus51198762

21199032

0

) (18)


119875eff = 0 (19)





is

(119875 +119886

V2) (V minus ) = 119896119879 (20)



119875eff = 119879eff1198614

21199031198881198612

+11986121198613minus 11986111198614

81205871199032119888119909 (1 + 119909) 119861

2

(21)


where

1198611=1 + 119909 minus 2119909

2+ 1199093+ 1199094

1 + 119909 + 1199092

1198612=1 + 119909 + 119909

2minus 21199093+ 1199094+ 1199095+ 1199096

1 + 119909 + 1199092

1198613=1 + 2119909 minus 2119909

4minus 1199095

(1 + 119909 + 1199092)2

1198614=1 + 2119909 + 3119909


(1 + 119909 + 1199092)2

(22)


V = 119903119888(1 minus 119909) (23)


120597119875eff120597V

= 01205972119875eff120597V2

= 0 (24)


(120597119875eff120597V)

119879eff

= minus1 times (81205871199031198885(1 + 119909) (1 + 119909 + 119909

2)3

times (1 + 119909 + 1199092minus 21199093+ 1199094+ 1199095+ 1199096) )

minus1


2minus 45119909

3minus 30119909

4minus 10119909

5

+ 61199096+ 71199097+ 41199098+ 1199099)

times 1198762119909 (15 + 54119909 + 98119909

2+ 104119909

3+ 59119909

4+ 18119909

5

minus 91199096minus 12119909

7minus 10119909

8minus 41199099minus 11990910)] = 0

(1205972119875eff120597V2

)

119879eff

= 1 times (41205871199031198886(1 + 119909) (1 + 119909 + 119909

2)4

times (1 + 119909 + 1199092minus 21199093+ 1199094+ 1199095+ 1199096)2

)

minus1

times [minus1199031198882119909 (6 + 36119909 + 115119909

2+ 286119909

3+ 486119909

4

+ 5301199095+ 203119909

6minus 357119909

7minus 783119909

8

minus 8541199099minus 666119909

10minus 375119909

11minus 127119909

12

+311990913+ 32119909

14+ 18119909

15+ 611990916+ 11990917)

+ 1198762(30 + 165119909 + 456119909

2+ 837119909

3+ 1050119909

4

+ 9031199095+ 255119909

6minus 720119909

7minus 1707119909

8

minus 21131199099minus 1743119909

10minus 969119909

11minus 314119909

12

minus 911990913+ 63119909

14+ 44119909

15+ 21119909

16+ 611990917

+ 11990918)] = 0

(25)



119909119888= 0871992 (26)


119876119888= 060395 V119888 = 0128008

119879119888= 000436559 119901

119888= 00000145468

(27)



119888=


119909119888= 0871992 (28)


119876119888= 301975 V119888 = 0640038

119879119888= 0000873119 119901

119888= 58187 times 10

minus7

(29)















119888increase



02 04 06 08 100000

0001

0002

0003

0004

0005

minus0001

minus0002

Peff


15

1

05

0

minus05

times10minus6

02 04 06 08 10

Peff

2

25

3












Acknowledgments


References


















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





Γ = 119890Δ119878++Δ119878119888 (7)




119878 = 119878++ 119878119888 (8)


119889119878 = 2120587(1

120581+

+1

120581119888

)119889119872

minus 2120587(120593+

120581+

+120593119888

120581119888

)119889119876 +120587

3(1199033

+

120581+

+1199033

119888

120581119888

)119889Λ

(9)


119889119878 = 2120587(1

120581+

+1

120581119888

)119889119872 +120587

3(1199033

+

120581+

+1199033

119888

120581119888

)119889Λ (10)


119889119903119888=119889119872

119903119888120581119888

+

(1199033

1198883)

2119903119888120581119888

119889Λ 119889119903+=119889119872

119903+120581+

+

(1199033

+3)

2119903+120581+

119889Λ

(11)


119881 =4120587

3(1199033

119888minus 1199033

+) (12)


119889119881 = 4120587(119903119888

120581119888

minus119903+

120581+

)119889119872 +2120587

3(1199034

119888

120581119888

minus1199034

+

120581+

)119889Λ (13)


119889119872 = 119879eff119889119878 minus 119875eff119889119881 (14)


119879eff =(1199094+ 1199093minus 21199092+ 119909 + 1)

4120587119903119888119909 (119909 + 1) (119909

2 + 119909 + 1)

minus1198762

412058711990331198881199093 (119909 + 1) (119909

2 + 119909 + 1)

times (1 + 119909 + 1199092minus 21199093+ 1199094+ 1199095+ 1199096)

(15)


119875eff =(1 minus 119909) (1 + 3119909 + 3119909

2+ 31199093+ 1199094)

81205871199032119888119909 (1 + 119909) (1 + 119909 + 119909

2)2

minus

1198762(1 minus 119909) (1 + 2119909 + 3119909


812058711990341198881199093 (1 + 119909) (1 minus 119909

3) (1 + 119909 + 1199092)

(16)

where 119909 = 119903+119903119888and 0 lt 119909 lt 1



1199032


2Λ= 1199032

0 (17)



+119888= 1199032

0 the


119879eff =1

121205871199030

(1 minus51198762

21199032

0

) (18)


119875eff = 0 (19)





is

(119875 +119886

V2) (V minus ) = 119896119879 (20)



119875eff = 119879eff1198614

21199031198881198612

+11986121198613minus 11986111198614

81205871199032119888119909 (1 + 119909) 119861

2

(21)


where

1198611=1 + 119909 minus 2119909

2+ 1199093+ 1199094

1 + 119909 + 1199092

1198612=1 + 119909 + 119909

2minus 21199093+ 1199094+ 1199095+ 1199096

1 + 119909 + 1199092

1198613=1 + 2119909 minus 2119909

4minus 1199095

(1 + 119909 + 1199092)2

1198614=1 + 2119909 + 3119909


(1 + 119909 + 1199092)2

(22)


V = 119903119888(1 minus 119909) (23)


120597119875eff120597V

= 01205972119875eff120597V2

= 0 (24)


(120597119875eff120597V)

119879eff

= minus1 times (81205871199031198885(1 + 119909) (1 + 119909 + 119909

2)3

times (1 + 119909 + 1199092minus 21199093+ 1199094+ 1199095+ 1199096) )

minus1


2minus 45119909

3minus 30119909

4minus 10119909

5

+ 61199096+ 71199097+ 41199098+ 1199099)

times 1198762119909 (15 + 54119909 + 98119909

2+ 104119909

3+ 59119909

4+ 18119909

5

minus 91199096minus 12119909

7minus 10119909

8minus 41199099minus 11990910)] = 0

(1205972119875eff120597V2

)

119879eff

= 1 times (41205871199031198886(1 + 119909) (1 + 119909 + 119909

2)4

times (1 + 119909 + 1199092minus 21199093+ 1199094+ 1199095+ 1199096)2

)

minus1

times [minus1199031198882119909 (6 + 36119909 + 115119909

2+ 286119909

3+ 486119909

4

+ 5301199095+ 203119909

6minus 357119909

7minus 783119909

8

minus 8541199099minus 666119909

10minus 375119909

11minus 127119909

12

+311990913+ 32119909

14+ 18119909

15+ 611990916+ 11990917)

+ 1198762(30 + 165119909 + 456119909

2+ 837119909

3+ 1050119909

4

+ 9031199095+ 255119909

6minus 720119909

7minus 1707119909

8

minus 21131199099minus 1743119909

10minus 969119909

11minus 314119909

12

minus 911990913+ 63119909

14+ 44119909

15+ 21119909

16+ 611990917

+ 11990918)] = 0

(25)



119909119888= 0871992 (26)


119876119888= 060395 V119888 = 0128008

119879119888= 000436559 119901

119888= 00000145468

(27)



119888=


119909119888= 0871992 (28)


119876119888= 301975 V119888 = 0640038

119879119888= 0000873119 119901

119888= 58187 times 10

minus7

(29)















119888increase



02 04 06 08 100000

0001

0002

0003

0004

0005

minus0001

minus0002

Peff


15

1

05

0

minus05

times10minus6

02 04 06 08 10

Peff

2

25

3












Acknowledgments


References


















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




where

1198611=1 + 119909 minus 2119909

2+ 1199093+ 1199094

1 + 119909 + 1199092

1198612=1 + 119909 + 119909

2minus 21199093+ 1199094+ 1199095+ 1199096

1 + 119909 + 1199092

1198613=1 + 2119909 minus 2119909

4minus 1199095

(1 + 119909 + 1199092)2

1198614=1 + 2119909 + 3119909


(1 + 119909 + 1199092)2

(22)


V = 119903119888(1 minus 119909) (23)


120597119875eff120597V

= 01205972119875eff120597V2

= 0 (24)


(120597119875eff120597V)

119879eff

= minus1 times (81205871199031198885(1 + 119909) (1 + 119909 + 119909

2)3

times (1 + 119909 + 1199092minus 21199093+ 1199094+ 1199095+ 1199096) )

minus1


2minus 45119909

3minus 30119909

4minus 10119909

5

+ 61199096+ 71199097+ 41199098+ 1199099)

times 1198762119909 (15 + 54119909 + 98119909

2+ 104119909

3+ 59119909

4+ 18119909

5

minus 91199096minus 12119909

7minus 10119909

8minus 41199099minus 11990910)] = 0

(1205972119875eff120597V2

)

119879eff

= 1 times (41205871199031198886(1 + 119909) (1 + 119909 + 119909

2)4

times (1 + 119909 + 1199092minus 21199093+ 1199094+ 1199095+ 1199096)2

)

minus1

times [minus1199031198882119909 (6 + 36119909 + 115119909

2+ 286119909

3+ 486119909

4

+ 5301199095+ 203119909

6minus 357119909

7minus 783119909

8

minus 8541199099minus 666119909

10minus 375119909

11minus 127119909

12

+311990913+ 32119909

14+ 18119909

15+ 611990916+ 11990917)

+ 1198762(30 + 165119909 + 456119909

2+ 837119909

3+ 1050119909

4

+ 9031199095+ 255119909

6minus 720119909

7minus 1707119909

8

minus 21131199099minus 1743119909

10minus 969119909

11minus 314119909

12

minus 911990913+ 63119909

14+ 44119909

15+ 21119909

16+ 611990917

+ 11990918)] = 0

(25)



119909119888= 0871992 (26)


119876119888= 060395 V119888 = 0128008

119879119888= 000436559 119901

119888= 00000145468

(27)



119888=


119909119888= 0871992 (28)


119876119888= 301975 V119888 = 0640038

119879119888= 0000873119 119901

119888= 58187 times 10

minus7

(29)















119888increase



02 04 06 08 100000

0001

0002

0003

0004

0005

minus0001

minus0002

Peff


15

1

05

0

minus05

times10minus6

02 04 06 08 10

Peff

2

25

3












Acknowledgments


References


















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




02 04 06 08 100000

0001

0002

0003

0004

0005

minus0001

minus0002

Peff


15

1

05

0

minus05

times10minus6

02 04 06 08 10

Peff

2

25

3












Acknowledgments


References


















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics










































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



research article the critical phenomena and thermodynamics...

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