Research Article Entropy Spectrum of a KS Black Hole in IR ...downloads.hindawi.com/journals/ahep/2014/396453.pdf · Research Article Entropy Spectrum of a KS Black Hole in IR Modified

Research ArticleEntropy Spectrum of a KS Black Hole in IR ModifiedHolava-Lifshitz Gravity

Shiwei Zhou12 Ge-Rui Chen1 and Yong-Chang Huang1

1 Institute of Theoretical Physics Beijing University of Technology Beijing 100124 China2Department of Foundation Academy of Armored Forces Engineering Beijing 100072 China

Correspondence should be addressed to Shiwei Zhou zhousw783163com

Received 6 January 2014 Accepted 10 February 2014 Published 13 March 2014

Academic Editor Xiaoxiong Zeng

Copyright copy 2014 Shiwei Zhou 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

As a renormalizable theory of gravity Horava-Lifshitz gravitymight be an ultraviolet completion of general relativity and reduces toEinstein gravitywith a nonvanishing cosmological constant in infrared Kehagias and Sfetsos obtained a static spherically symmetricblack hole solution called KS black hole in the IR modified Horava-Lifshitz theory In this paper the entropy spectrum and areaspectrum of a KS black hole are investigated based on the proposal of adiabatic invariant quantity By calculating the action ofproducing a pair of particles near the horizon it is obtained that the action of the system is exactly equivalent to the change of blackhole entropy which is an adiabatic invariant quantity With the help of Bohr-Sommerfeld quantization rule it is concluded thatthe entropy spectrum is discrete and equidistant spaced and the area spectrum is not equidistant spaced which depends on theparameter of gravity theory Some summary and discussion will be given in the last

1 Introduction

In the 1970rsquos with the discovery of Hawking radiationand Bekensteinrsquos proposal of black hole entropy black holethermodynamics has been built up successfully which hasopened a new field to study quantum theory and gravitytheory [1ndash4] Nowadays there has been some trouble on thestatistic origin and quantization of black hole entropy forphysicists on black hole thermodynamics Since Bekensteinproposed that the horizon area of a nonextremal black holeis an adiabatic invariant classically and the horizon areaof black hole is quantized in units of 1198972

119901[5ndash8] there has

been much attention paid to the quantization of black holeentropy spectrum and area spectrum Hod proposed that ifone employs Bohrrsquos corresponding principle the real part ofthe quasinormal mode frequency is responsible for the areaspectrum of black hole [9 10] Combining the proposal byBekenstein for the adiabaticity of black hole horizon areaandHodrsquos proposal Kunstatter proved that entropy spectrumof a 119889-dimensional black hole is quantized and the result isin agreement with that of Hod and Bekenstein [11] Later

Maggiore gave a new interpretation of black hole quasinormalmodes in connection to the quantization of black holehorizon area An important statement in Maggiorersquos work isthat the periodicity of a black hole in Euclidean time maybe the origin of area quantization [12] It is well known thatfor any background spacetime with a horizon in Kruskalcoordinates the period with respect to Euclidean time takesthe form of 119879 = 2120587120581 where 120581 is the surface gravityof the horizon Vagenas exclusively used the fact that theblack hole horizon area is an adiabatic invariant quantumand derived an equally spaced entropy spectrum of a blackhole with its value to be equal to that of Bekenstein [13]Zeng et al considered that the action 119868 action variable 119868Vand Hamiltonian119867 of any single periodic system satisfy therelation 119868 = 119868V minus int119867119889119905 They proposed that the actionvariable can be quantized with the equally spaced form119868V = 2120587119899ℎ Once the action and Hamiltonian are given thequantization action variable can be obtained With the helpof Bohr-Sommerfeld quantization rule they proved that thequantized action variable is nothing but the entropy of blackhole thus the entropy and the horizon area of a black hole

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2014 Article ID 396453 6 pageshttpdxdoiorg1011552014396453

2 Advances in High Energy Physics

can be quantized They emphasized that the action variableshould be adiabatic invariant [14ndash16] Recently Jiang andHanargued that the adiabatic invariant quantity int119901

119894119889119902119894is not

canonically invariant and the adiabatic invariant quantumshould be of the covariance form ∮119901

119894119889119902119894= 119899ℎ They also

obtained the equally spaced entropy spectrum with the formof Δ119878 = 2120587 [17] Some more works on the quantization ofentropy spectrum of a black hole can be seen in the paper[18ndash23]

Horava gravity is a nonrelativistic renormalizable theoryof gravitation [24] which is inspired by the anisotropicscaling between time and space in condensed matter systemsin particular in the theory of quantum critical phenomenawhere the degree of anisotropy between space and timeis characterized by the ldquodynamical critical exponentrdquo 119911 Itis well known that relativistic systems automatically satisfy119911 = 1 as a consequence of Lorentz invariance In Horava-Lifshitz theory systemsrsquo scaling at a short distance exhibits astrong anisotropy between space and time with 119911 gt 1 Thiswill improve the short-distance behavior of the theory Theanisotropy at short distance can be lost for long distancewhilethe Lorentz symmetry will appear as an emergent symmetryThe black hole solution of original Horava-Lifshitz gravitydoes not recover the usual Schwarzschild-anti-de Sitter blackhole with the detailed-balance condition A relevant operatorproportional to the 3119863 geometry Ricci scalar of the originalHorava-Lifshitz theory action was introduced [25] and itdeviated from detailed balance This does not modify theultraviolet properties of the theory However it modifies theinfrared Horava-Lifshitz gravity theory So a Schwarzschild-anti-de Sitter solution can be realized in infrared modifiedHorava-Lifshitz gravity theory and the Minkowski vacuumis also allowed On the limit of vanishing Λ

120596 a spherically

symmetric black hole solution has been obtained by Kehagiasand Sfetsos [25] which is the analogy of Schwarzschild blackhole in general relativity and is exactly asymptotically flatAfter Horava-Lifshitz gravity was proposed much attentionhas been paid to it [26ndash31]

In this paper based on Vagenasrsquo proposal of adiabaticinvariant quantity we investigate entropy spectrum of a KSblack hole in IR modified Horava-Lifshitz gravity Vagenaspointed that the general coordinates 119902

119894contain 119902

0= 120591 and

1199021= 119903 and the contribution of the two parts to the adiabatic

invariant quantity are equivalent with each other by defining119903 = 119889119903119889120591 Thus the integration result of adiabatic invariantquantity was directly given by the 120591-integration where theperiod 119879 of gravity system satisfies 119879 = 2120587120581 We find thatthe periodicity of gravity system can conveniently be used tocalculate the entropy spectrumHowever the physical pictureof periodicity is not clear We give our explanation about it byconsidering a process that a pair of particles create outside thehorizon One period of the system corresponds to the processwith the outgoing positive energy particle crossing outwardsthe horizon while the negative energy particle tunnels intothe black hole The movement of the two particles can bedescribed as a tunneling process proposed by Parikh andWilczekrsquos Hawking radiation [32] After calculating we findthat the action of the system is exactly the black hole entropy

which is the adiabatic invariant quantity With the help ofBohr-Sommerfeld quantization rule we obtain the quantizedentropy and area spectrum Some summary and discussionwill be given in the latter

2 Review of a KS Black Hole in IR ModifiedHolava-Lifshitz Gravity

Using the ADM decomposition of the metric

1198891199042= minus119873

21198891199052+ 119892119894119895(119889119909119894+ 119873119894119889119905) (119889119909

119895+ 119873119895119889119905) (1)

where119873 and119873119894 are the ldquolapserdquo and ldquoshiftrdquo variables respec-tively and 119892

119894119895is the spatial metric On the limit of Λ

120596rarr 0

the action of IR modified Horava-Lifshitz gravity theory canbe given by

119878 = int1198891199051198893119909radic119892119873

times [2

1205812(119870119894119895119870119894119895minus 1205821198702)

minus1205812

21205964119862119894119895119862119894119895+1205812120583

21205962120598119894119895119896119877(3)

119894119897nabla119895119877(3)119897

119896

minus12058121205832

8119877(3)

119894119895119877(3)119894119895

+12058121205832

8 (1 minus 3120582)

1 minus 4120582

4(119877(3))2

+ 1205834119877(3)]

(2)

which is obtained by introducing a term proportional to theRicci scalar of the three-geometry1205834119877(3) to the original actionof Horava-Lifshitz gravity [26 27] Here 119870

119894119895is the extrinsic

curvature defined by

119870119894119895=

1

2119873( 119892119894119895minus nabla119894119873119895minus nabla119895119873119894) (3)

where the dot denotes a derivative with respect to 119905 119862119894119895 is theCotton tensor defined by

119862119894119895= 120598119894119896119897nabla119896(119877(3)119895

119897minus1

4119877(3)120575119895

119897) (4)

and 119877(3) is the 3-dimensional curvature scalar for 119892

119894119895

120581 120582 120596 120583 are all coupling constant parametersComparing the action for the case of 120582 = 1 with the

standard Einstein-Hilbert action we find that the Lagrangianwill become the usual Einstein-Hilbert Lagrangian when thespeed of light 119888 Newtonrsquos constant 119866 and the cosmologicalconstant Λ are given by

1198882=12058121205834

2 119866 =

1205812

32120587119888 Λ =

3

2Λ119882 (5)

The spherically symmetric asymptotically flat black holesolution has been obtained by Cai et al [27] which is the

Advances in High Energy Physics 3

analogy of Schwarzschild black hole in general relativity Themetric can be written as

1198891199042= minus119891 (119903) 119889119905

2+

1

119891 (119903)1198891199032+ 1199032(1198891205792+ sin21205791198891205932) (6)

with

119891 (119903) = 1 + 1205961199032minus radic119903 (12059621199033 + 4120596119872) (7)

where119872 is an integration constant corresponding to themassof black hole and 120596 is a coupling constant parameter

The condition 119891(119903plusmn) = 0 gives the outer and inner

horizons at

119903plusmn= 119872(1 plusmn radic1 minus

1

21205961198722) (8)

To avoid naked singularity we should have1205961198722 ge 12 Inthe regime of traditional general relativity we have 1205961198722 ≫1 so the outer horizon approaches the usual Schwarzschildhorizon 119903

+≃ 2119872 whereas the inner one approaches the

singularity 119903minus≃ 0

3 Entropy Quantization via AdiabaticInvariant Action

We consider a process that a pair of particles create near thehorizonWhile the outgoing positive energy particle crossingoutwards the horizon the negative energy particle ingoingtowards the black hole along the radial directionWe describethe movement of the two particles as a tunneling processproposed by Parikh and Wilczekrsquos proposal [32]

The action of the system is

119868 = int119901119903119889119903 = int

119903out

119903in

119901119903119889119903 + int

119903in

119903out

119901119903119889119903

= int

119903out

119903in

int

119901119903

0

1198891199011015840

119903119889119903 + int

119903in

119903out

int

119901119903

0

1198891199011015840

119903119889119903

(9)

The first term is corresponding to the particles with positiveenergy and the second term is the negative energy oneWhenenergy conservation is considered the black hole mass willdecrease with the outgoing particle emitting The Hamilton119867 ADM energy 119872 and the particlersquos energy 1198981015840 satisfy therelation 119867 = 119872 minus 119898

1015840 that is 119889119867 = minus1198891198981015840 Then by use

of Hamiltonrsquos equation 119903 = 119889119867119889119901119903 the first term of (9)

becomes

1198681equiv int

119903out

119903in

int

119901119903

0

1198891199011015840

119903119889119903 = int

119903out

119903in

int

119872minus119898

119872

119889119867

119903119889119903

= int

119898

0

int

119903out

119903in

119889119903

119903(minus119889119898

1015840)

(10)

where 119903in = 119903ℎ(119872) minus 120598 119903out = 1199031015840

ℎ(119872 minus 119898) + 120598 for the reason

of that the black hole horizon will decrease with the particleemitting out

When considering 119903 = 119889119903119889120591 = 119891(119903) we get

1198681= int

119898

0

int

119903out

119903in

119889119903

1 + 1205961199032 minus 119903 [12059621199033 + 4120596 (119872 minus 1198981015840)]1198891198981015840 (11)

It is easily found that there is a pole at 1199031015840ℎ= (119872 minus 119898

1015840) +

radic(119872 minus 1198981015840)2minus 12120596 We do the integration as follows

1198681= int

119898

0

int

119903out

119903in

(119889119903((119903 minus 1199031015840

ℎ)

times[[

[

2120596119903

minus212059621199033+ 2120596 (119872minus119898

1015840)

radic119903 (12059621199033+4120596 (119872minus1198981015840))

]]

]

)

minus1

)1198891198981015840

= 2120587int

119898

0

(1(21205961199031015840

ℎ

minus2120596211990310158403

ℎ+ 2120596 (119872 minus 119898

1015840)

radic1199031015840ℎ[120596211990310158403ℎ+ 4120596 (119872 minus 1198981015840)]

)

minus1

)1198891198981015840

= 2120587int

119898

0

((1

2+ 2120596(119872 minus 119898

1015840)2

+ 2120596 (119872 minus 1198981015840)radic(119872 minus 1198981015840)

2

minus1

2120596)

times(2120596radic(119872 minus 1198981015840)2

minus1

2120596)

minus1

)1198891198981015840

= 2120587[[

[

int

119898

0

1

4120596radic(119872 minus 1198981015840)2

minus 12120596

1198891198981015840

+ int

119898

0

(119872 minus 1198981015840)2

radic(119872 minus 1198981015840)2

minus 12120596

1198891198981015840

+int

119898

0

(119872 minus 1198981015840) 1198891198981015840]]

]


= 2120587[119872

2radic1198722 minus

1

2120596minus119872 minus 119898

1015840

2radic(119872 minus 1198981015840)

2

minus1

2120596

+1

2120596[ln119872+radic1198722 minus

1

2120596minus ln (119872 minus 119898)

+radic(119872 minus 119898)2minus1

2120596] +119872119898 minus

1198982

2]

(12)

The second term of (9) corresponds to ingoing particleswith negative energy After similar calculation as the firstterm we find that the contribution is equivalent to that of 119868

1

That is

119868 = 21198681 (13)

On the other hand the Hawking temperature of the outerevent horizon has been obtained as [31]

119879119867=

21205961199032

+minus 1

8120587 (1205961199033++ 119903+)

=radic1198722 minus 12120596

2120587 (21198722 + 2119872radic1198722 minus 12120596 + 12120596)

(14)

which is proportional to the surface gravity 120581 =

(12)(120597119891(119903)120597119903)|119903=119903+

on the event horizon of black holeConsidering the first law of thermodynamics and substitutingthe expression of temperature of black hole we can obtainthe entropy

119878BH =119860

4+120587

120596ln 1198604 (15)

where 119860 = 1205871199032+is the area of event horizon

After calculation we find that the change of black holeentropy Δ119878 when a particle with energy of119898 emits out of theblack hole is exactly equivalent to the action 119868 that is

Δ119878 = 120587 [1199032

+(119872) minus 119903

2

+(119872 minus 119898)] = 119868 (16)

Now using the periodicity of the black hole we calculatethe adiabatic invariant quantity According to the dimen-sional reduction technique the two-dimensional spacetimeof a KS black hole can be given by

1198891199042= minus119891 (119903) 119889119905

2+ 119891(119903)

minus11198891199032 (17)

When defining the tortoise coordinate as

119903lowast= 119903 +

1

2120581+

ln119903 minus 119903+

119903+

+1

2120581minus

ln119903 minus 119903minus

119903minus

(18)

in which 120581plusmn= (1198911015840(119903)2)|

119903=119903plusmn

= (21205961199032

plusmnminus1)(4119903

plusmn(1205961199032

plusmn+1)) is the

surface gravity on the outer (inner) horizon Using the null

coordinates 119906 = 119905 minus 119903lowast V = 119905 + 119903

lowast we can get the coordinates

119880 = minus119890minus120581+119906 and 119881 = 119890120581+V [33 34] Then define

119879 =1

2(119881 + 119880)

= 119890120581+119903(119903 minus 119903+

119903+

)

12

(119903 minus 119903minus

119903minus

)

120581+2120581minus

sinh 120581+119905

119877 =1

2(119881 minus 119880) = 119890

120581+119903(119903 minus 119903+

119903+

)

12


119903minus

)

120581+2120581minus

cosh 120581+119905

(19)

where 119879 119877 are the Kruskal-like coordinatesDifferent from (18) that is

119889119903lowast= 119891(119903)

minus1119889119903 (20)

the two-dimensional KS metric becomes

1198891199042= minus119889119905

2+ 1198891199032

lowast

= 120581minus2

+119890minus2120581+119903(119903 minus 119903minus

119903+

)(119903minus

119903 minus 119903minus

)

120581+120581minus

times [minus1198891198792+ 1198891198772]

(21)

Transforming the time coordinate as 119905 rarr minus119894120591 we have

119894119879 = 119890120581+119903(119903 minus 119903+

119903+

)

12


119903minus

)

120581+2120581minus

sin 120581+120591

119877 = 119890120581+119903(119903 minus 119903+

119903+

)

12


119903minus

)

120581+2120581minus

cos 120581+120591

(22)

It is easily found that both 119879 119877 are periodic functionswith respect to the Euclidean time 120591 with the period of2120587120581+ Since we only consider the case of outer horizon for

simplicity we write 120581+for 120581 from now on

When utilizing Vagenasrsquos adiabatic invariant quantitywith the following form

119868 = int119901119894119889119902119894= intint

119901119894

0

1198891199011015840

119894119889119902119894= intint

119867

0

1198891198671015840

119902119894

119889119902119894

= intint

119867

0

1198891198671015840119889120591 + intint

119867

0

1198891198671015840

119903119889119903

(23)

where 119901119894are the conjugate momentum of the general coor-

dinate 119902119894with 119894 = 0 1 for 119902

0= 120591 and 119902

1= 119903 Considering

119903 = 119889119903119889120591 we have

119868 = 2intint

119867

0

1198891198671015840119889120591 (24)

Because of the periodicity of 120591 with 119879 = 2120587ℎ120581 the adiabaticinvariant quantity can be calculated as

119868 = 2120587int

119867

0

1198891198671015840

120581= ℎint

119867

0

1198891198671015840

119879BH= ℎ119878BH (25)


Implementing the Bohr-Sommerfeld quantization condi-tion

∮119901119889119902 = 2120587119899ℎ (26)

the black hole entropy spectrum can be given as

119878BH = 2120587119899 119899 = 1 2 3 (27)

and the entropy spectrum is discrete and equidistant spacedwith

Δ119878BH = 2120587 (28)

To get the area spectrum differentiate (15)

Δ119878BH =1

4Δ119860 +

120587

120596

4

119860Δ119860 (29)

we have

Δ119860 =Δ119878BH

14 + 4120587119860120596≃ 8120587(1 minus

4

1205961199032+

) (30)

We find that area spectrum is not equidistant spaced

4 Summary and Conclusion

In this paper based on the idea of adiabatic invariantquantity we have investigated entropy spectrum of a KSblack hole in IR modified Horava-Lifshitz gravity As amodified gravity theory the entropy of a KS black holedoes not satisfy Bekensteinrsquos entropy-area relation It consistsof two terms one is the Bekenstein-Hawking entropy theother is a logarithmic term The discrepancy between theentropy and the Bekenstein-Hawking entropy is the reflectionof differences between this modified gravity theory andgeneral relativity After calculating we find that the blackhole entropy is an adiabatic invariant quantity With thehelp of Bohr-Sommerfeld quantization rule we obtain thequantized entropy and area spectrum It is concluded thatthe entropy spectrum can be given as 119878BH = 2120587119899 with119899 = 1 2 3 which is discrete and equidistant spacedwith Δ119878BH = 2120587 and the area spectrum is not equidistantspaced which depends on the parameter of gravity theoryIn addition by calculating the action of a production of apair of particles near the horizon we find that the actionof the system is exactly equivalent to the change to blackhole entropy which is an adiabatic invariant quantity Theprocession of the particle producing with positive energyoutgoing towards the horizon while the one with negativeenergy is ingoing the horizon can give a clear explanation tothe periodicity of gravity system

Conflict of Interests

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

Acknowledgments

This work is supported by Beijing Postdoctoral ResearchFoundation no 71006015201201 andNational Natural ScienceFoundation of China (no 11275017 and no 11173028)

References

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

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

[3] S W Hawking ldquoInformation loss in black holesrdquo PhysicalReview D vol 72 no 8 Article ID 084013 4 pages 2005

[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] J D Bebenstein ldquoBlack holes and entropyrdquo Physical Review Dvol 7 no 8 pp 2333ndash2346 1973

[6] J D Bebenstein ldquoBlack holes and the second lawrdquo Lettere AlNuovo Cimento vol 4 pp 737ndash740 1972

[7] J D Bebenstein ldquoThequantummass spectrumof theKerr blackholerdquo Lettere Al Nuovo Cimento vol 11 pp 467ndash470 1974

[8] J D Bebenstein ldquoExtraction of energy and charge from a blackholerdquo Physical Review D vol 7 pp 949ndash953 1973

[9] SHod ldquoBohrrsquos correspondence principle and the area spectrumof quantum black holesrdquo Physical Review Letters vol 81 no 20pp 4293ndash4296 1998

[10] S Hod ldquoGravitation the quantum and Bohrrsquos correspondenceprinciplerdquo General Relativity and Gravitation vol 31 no 11 pp1639ndash1644 1999

[11] G Kunstatter ldquo119889-dimensional black hole entropy spectrumfrom quasinormal modesrdquo Physical Review Letters vol 90 no16 Article ID 161301 4 pages 2003

[12] M Maggiore ldquoPhysical interpretation of the spectrum of blackhole quasinormal modesrdquo Physical Review Letters vol 100 no14 Article ID 141301 4 pages 2008

[13] B R Majhi and E C Vagenas ldquoBlack hole spectroscopy viaadiabatic invariancerdquo Physics Letters B vol 701 pp 623ndash6252011

[14] X X Zeng X M Liu and W B Liu ldquoPeriodicity and areaspectrum of black holesrdquoThe European Physical Journal C vol72 p 1967 2012

[15] X X Zeng and W B Liu ldquoSpectroscopy of a Reissner-Nordstrom black hole via an action variablerdquo The EuropeanPhysical Journal C vol 72 p 1987 2012

[16] X-M Liu X-X Zeng and S-W Zhou ldquoArea spectra of BTZblack holes via periodicityrdquo Science China Physics Mechanicsand Astronomy vol 56 no 9 pp 1627ndash1631 2013

[17] Q-Q Jiang and Y Han ldquoOn black hole spectroscopy viaadiabatic invariancerdquo httparxivorgabs12104002

[18] H-L Li R Lin and L-Y Chen ldquoEntropy quantization ofReissner-Nordstrom de Sitter black hole via adiabatic covariantactionrdquoGeneral Relativity and Gravitation vol 45 pp 865ndash8752013

[19] D Chen andHYang ldquoEntropy spectrumof aKerr anti-de Sitterblack holerdquo The European Physical Journal C vol 72 p 20272012

[20] R Tharanath and V C Kuriakose ldquoThermodynamics andspectroscopy of charged dilaton black holesrdquo General Relativityand Gravitation vol 45 no 9 pp 1761ndash1770 2013


[21] Q Q Jiang ldquoRevisit emission spectrum and entropy quantumof the Reissner-Nordstrom black holerdquo The European PhysicalJournal C vol 72 p 2086 2012

[22] C-Z Liu ldquoBlack hole spectroscopy via adiabatic invariant in aquantum corrected spacetimerdquo The European Physical JournalC vol 72 p 2009 2012

[23] T Tanaka and T Tamaki ldquoArea spectrum of horizon and blackhole entropyrdquoThe European Physical Journal C vol 73 p 23142013

[24] PHorava ldquoQuantumgravity at a Lifshitz pointrdquoPhysical ReviewD vol 79 no 8 Article ID 084008 15 pages 2009

[25] A Kehagias and K Sfetsos ldquoThe black hole and FRW geome-tries of non-relativistic gravityrdquo Physics Letters B vol 678 no 1pp 123ndash126 2009

[26] H Lu J Mei and C N Pope ldquoSolutions to horava gravityrdquoPhysical Review Letters vol 103 no 9 Article ID 091301 2009

[27] R-G Cai L-M Cao and N Ohta ldquoTopological black holesin Horava-Lifshitz gravityrdquo Physical Review D vol 80 no 2Article ID 024003 7 pages 2009

[28] S Mukohyama ldquoScale-invariant cosmological perturbationsfrom Horava-Lifshitz gravity without inflationrdquo Journal ofCosmology and Astroparticle Physics vol 2009 article 001 2009

[29] S Mukohyama K Nakayama F Takahashi and S YokoyamaldquoPhenomenological aspects of Horava-Lifshitz cosmologyrdquoPhysics Letters B vol 679 no 1 pp 6ndash9 2009

[30] S-W Zhou and W-B Liu ldquoThree classical tests of Horava-Lifshitz gravity theoryrdquo Astrophysics and Space Science vol 337no 2 pp 779ndash784 2012

[31] S-W Zhou and W-B Liu ldquoBlack hole thermodynamics ofHorava-Lifshitz and IR modified Horava-Lifshitz gravity the-oryrdquo International Journal of Theoretical Physics vol 50 no 6pp 1776ndash1784 2011

[32] M K Parikh and F Wilczek ldquoHawking radiation as tunnelingrdquoPhysical Review Letters vol 85 no 24 pp 5042ndash5045 2000

[33] C Liang and B Zhou Differential Geometry and GeneralRelativity Science Press Beijing China 2006

[34] Z ZhaoTheThermal Nature of Black Holes and the Singularityof the Space-Time Beijing Normal University Press BeijingChina 1999

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can be quantized They emphasized that the action variableshould be adiabatic invariant [14ndash16] Recently Jiang andHanargued that the adiabatic invariant quantity int119901

119894119889119902119894is not

canonically invariant and the adiabatic invariant quantumshould be of the covariance form ∮119901

119894119889119902119894= 119899ℎ They also

obtained the equally spaced entropy spectrum with the formof Δ119878 = 2120587 [17] Some more works on the quantization ofentropy spectrum of a black hole can be seen in the paper[18ndash23]

Horava gravity is a nonrelativistic renormalizable theoryof gravitation [24] which is inspired by the anisotropicscaling between time and space in condensed matter systemsin particular in the theory of quantum critical phenomenawhere the degree of anisotropy between space and timeis characterized by the ldquodynamical critical exponentrdquo 119911 Itis well known that relativistic systems automatically satisfy119911 = 1 as a consequence of Lorentz invariance In Horava-Lifshitz theory systemsrsquo scaling at a short distance exhibits astrong anisotropy between space and time with 119911 gt 1 Thiswill improve the short-distance behavior of the theory Theanisotropy at short distance can be lost for long distancewhilethe Lorentz symmetry will appear as an emergent symmetryThe black hole solution of original Horava-Lifshitz gravitydoes not recover the usual Schwarzschild-anti-de Sitter blackhole with the detailed-balance condition A relevant operatorproportional to the 3119863 geometry Ricci scalar of the originalHorava-Lifshitz theory action was introduced [25] and itdeviated from detailed balance This does not modify theultraviolet properties of the theory However it modifies theinfrared Horava-Lifshitz gravity theory So a Schwarzschild-anti-de Sitter solution can be realized in infrared modifiedHorava-Lifshitz gravity theory and the Minkowski vacuumis also allowed On the limit of vanishing Λ

120596 a spherically

symmetric black hole solution has been obtained by Kehagiasand Sfetsos [25] which is the analogy of Schwarzschild blackhole in general relativity and is exactly asymptotically flatAfter Horava-Lifshitz gravity was proposed much attentionhas been paid to it [26ndash31]

In this paper based on Vagenasrsquo proposal of adiabaticinvariant quantity we investigate entropy spectrum of a KSblack hole in IR modified Horava-Lifshitz gravity Vagenaspointed that the general coordinates 119902

119894contain 119902

0= 120591 and

1199021= 119903 and the contribution of the two parts to the adiabatic

invariant quantity are equivalent with each other by defining119903 = 119889119903119889120591 Thus the integration result of adiabatic invariantquantity was directly given by the 120591-integration where theperiod 119879 of gravity system satisfies 119879 = 2120587120581 We find thatthe periodicity of gravity system can conveniently be used tocalculate the entropy spectrumHowever the physical pictureof periodicity is not clear We give our explanation about it byconsidering a process that a pair of particles create outside thehorizon One period of the system corresponds to the processwith the outgoing positive energy particle crossing outwardsthe horizon while the negative energy particle tunnels intothe black hole The movement of the two particles can bedescribed as a tunneling process proposed by Parikh andWilczekrsquos Hawking radiation [32] After calculating we findthat the action of the system is exactly the black hole entropy

which is the adiabatic invariant quantity With the help ofBohr-Sommerfeld quantization rule we obtain the quantizedentropy and area spectrum Some summary and discussionwill be given in the latter

2 Review of a KS Black Hole in IR ModifiedHolava-Lifshitz Gravity

Using the ADM decomposition of the metric

1198891199042= minus119873

21198891199052+ 119892119894119895(119889119909119894+ 119873119894119889119905) (119889119909

119895+ 119873119895119889119905) (1)

where119873 and119873119894 are the ldquolapserdquo and ldquoshiftrdquo variables respec-tively and 119892

119894119895is the spatial metric On the limit of Λ

120596rarr 0

the action of IR modified Horava-Lifshitz gravity theory canbe given by

119878 = int1198891199051198893119909radic119892119873

times [2

1205812(119870119894119895119870119894119895minus 1205821198702)

minus1205812

21205964119862119894119895119862119894119895+1205812120583

21205962120598119894119895119896119877(3)

119894119897nabla119895119877(3)119897

119896

minus12058121205832

8119877(3)

119894119895119877(3)119894119895

+12058121205832

8 (1 minus 3120582)

1 minus 4120582

4(119877(3))2

+ 1205834119877(3)]

(2)

which is obtained by introducing a term proportional to theRicci scalar of the three-geometry1205834119877(3) to the original actionof Horava-Lifshitz gravity [26 27] Here 119870

119894119895is the extrinsic

curvature defined by

119870119894119895=

1

2119873( 119892119894119895minus nabla119894119873119895minus nabla119895119873119894) (3)

where the dot denotes a derivative with respect to 119905 119862119894119895 is theCotton tensor defined by

119862119894119895= 120598119894119896119897nabla119896(119877(3)119895

119897minus1

4119877(3)120575119895

119897) (4)

and 119877(3) is the 3-dimensional curvature scalar for 119892

119894119895

120581 120582 120596 120583 are all coupling constant parametersComparing the action for the case of 120582 = 1 with the

standard Einstein-Hilbert action we find that the Lagrangianwill become the usual Einstein-Hilbert Lagrangian when thespeed of light 119888 Newtonrsquos constant 119866 and the cosmologicalconstant Λ are given by

1198882=12058121205834

2 119866 =

1205812

32120587119888 Λ =

3

2Λ119882 (5)

The spherically symmetric asymptotically flat black holesolution has been obtained by Cai et al [27] which is the



1198891199042= minus119891 (119903) 119889119905

2+

1

119891 (119903)1198891199032+ 1199032(1198891205792+ sin21205791198891205932) (6)

with

119891 (119903) = 1 + 1205961199032minus radic119903 (12059621199033 + 4120596119872) (7)



horizons at


1

21205961198722) (8)







119868 = int119901119903119889119903 = int

119903out

119903in

119901119903119889119903 + int

119903in

119903out

119901119903119889119903

= int

119903out

119903in

int

119901119903

0

1198891199011015840

119903119889119903 + int

119903in

119903out

int

119901119903

0

1198891199011015840

119903119889119903

(9)




becomes

1198681equiv int

119903out

119903in

int

119901119903

0

1198891199011015840

119903119889119903 = int

119903out

119903in

int

119872minus119898

119872

119889119867

119903119889119903

= int

119898

0

int

119903out

119903in

119889119903

119903(minus119889119898

1015840)

(10)





1198681= int

119898

0

int

119903out

119903in

119889119903

1 + 1205961199032 minus 119903 [12059621199033 + 4120596 (119872 minus 1198981015840)]1198891198981015840 (11)


1015840) +


1198681= int

119898

0

int

119903out

119903in

(119889119903((119903 minus 1199031015840

ℎ)

times[[

[

2120596119903

minus212059621199033+ 2120596 (119872minus119898

1015840)

radic119903 (12059621199033+4120596 (119872minus1198981015840))

]]

]

)

minus1

)1198891198981015840

= 2120587int

119898

0

(1(21205961199031015840

ℎ

minus2120596211990310158403

ℎ+ 2120596 (119872 minus 119898

1015840)

radic1199031015840ℎ[120596211990310158403ℎ+ 4120596 (119872 minus 1198981015840)]

)

minus1

)1198891198981015840

= 2120587int

119898

0

((1

2+ 2120596(119872 minus 119898

1015840)2


2

minus1

2120596)


minus1

2120596)

minus1

)1198891198981015840

= 2120587[[

[

int

119898

0

1

4120596radic(119872 minus 1198981015840)2

minus 12120596

1198891198981015840

+ int

119898

0

(119872 minus 1198981015840)2

radic(119872 minus 1198981015840)2

minus 12120596

1198891198981015840

+int

119898

0

(119872 minus 1198981015840) 1198891198981015840]]

]


= 2120587[119872

2radic1198722 minus

1

2120596minus119872 minus 119898

1015840

2radic(119872 minus 1198981015840)

2

minus1

2120596

+1

2120596[ln119872+radic1198722 minus

1

2120596minus ln (119872 minus 119898)


2120596] +119872119898 minus

1198982

2]

(12)


1

That is

119868 = 21198681 (13)


119879119867=

21205961199032

+minus 1

8120587 (1205961199033++ 119903+)

=radic1198722 minus 12120596

2120587 (21198722 + 2119872radic1198722 minus 12120596 + 12120596)

(14)


(12)(120597119891(119903)120597119903)|119903=119903+


119878BH =119860

4+120587

120596ln 1198604 (15)



Δ119878 = 120587 [1199032

+(119872) minus 119903

2

+(119872 minus 119898)] = 119868 (16)


1198891199042= minus119891 (119903) 119889119905

2+ 119891(119903)

minus11198891199032 (17)


119903lowast= 119903 +

1

2120581+

ln119903 minus 119903+

119903+

+1

2120581minus


119903minus

(18)

in which 120581plusmn= (1198911015840(119903)2)|

119903=119903plusmn

= (21205961199032


plusmn(1205961199032

plusmn+1)) is the





119879 =1

2(119881 + 119880)

= 119890120581+119903(119903 minus 119903+

119903+

)

12


119903minus

)

120581+2120581minus

sinh 120581+119905

119877 =1

2(119881 minus 119880) = 119890

120581+119903(119903 minus 119903+

119903+

)

12


119903minus

)

120581+2120581minus

cosh 120581+119905

(19)


119889119903lowast= 119891(119903)

minus1119889119903 (20)


1198891199042= minus119889119905

2+ 1198891199032

lowast

= 120581minus2


119903+

)(119903minus


)

120581+120581minus

times [minus1198891198792+ 1198891198772]

(21)


119894119879 = 119890120581+119903(119903 minus 119903+

119903+

)

12


119903minus

)

120581+2120581minus

sin 120581+120591

119877 = 119890120581+119903(119903 minus 119903+

119903+

)

12


119903minus

)

120581+2120581minus

cos 120581+120591

(22)




119868 = int119901119894119889119902119894= intint

119901119894

0

1198891199011015840

119894119889119902119894= intint

119867

0

1198891198671015840

119902119894

119889119902119894

= intint

119867

0

1198891198671015840119889120591 + intint

119867

0

1198891198671015840

119903119889119903

(23)


dinate 119902119894with 119894 = 0 1 for 119902

0= 120591 and 119902


119903 = 119889119903119889120591 we have

119868 = 2intint

119867

0

1198891198671015840119889120591 (24)


119868 = 2120587int

119867

0

1198891198671015840

120581= ℎint

119867

0

1198891198671015840

119879BH= ℎ119878BH (25)



∮119901119889119902 = 2120587119899ℎ (26)


119878BH = 2120587119899 119899 = 1 2 3 (27)


Δ119878BH = 2120587 (28)


Δ119878BH =1

4Δ119860 +

120587

120596

4

119860Δ119860 (29)

we have

Δ119860 =Δ119878BH

14 + 4120587119860120596≃ 8120587(1 minus

4

1205961199032+

) (30)






Acknowledgments


References









































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





1198891199042= minus119891 (119903) 119889119905

2+

1

119891 (119903)1198891199032+ 1199032(1198891205792+ sin21205791198891205932) (6)

with

119891 (119903) = 1 + 1205961199032minus radic119903 (12059621199033 + 4120596119872) (7)



horizons at


1

21205961198722) (8)







119868 = int119901119903119889119903 = int

119903out

119903in

119901119903119889119903 + int

119903in

119903out

119901119903119889119903

= int

119903out

119903in

int

119901119903

0

1198891199011015840

119903119889119903 + int

119903in

119903out

int

119901119903

0

1198891199011015840

119903119889119903

(9)




becomes

1198681equiv int

119903out

119903in

int

119901119903

0

1198891199011015840

119903119889119903 = int

119903out

119903in

int

119872minus119898

119872

119889119867

119903119889119903

= int

119898

0

int

119903out

119903in

119889119903

119903(minus119889119898

1015840)

(10)





1198681= int

119898

0

int

119903out

119903in

119889119903

1 + 1205961199032 minus 119903 [12059621199033 + 4120596 (119872 minus 1198981015840)]1198891198981015840 (11)


1015840) +


1198681= int

119898

0

int

119903out

119903in

(119889119903((119903 minus 1199031015840

ℎ)

times[[

[

2120596119903

minus212059621199033+ 2120596 (119872minus119898

1015840)

radic119903 (12059621199033+4120596 (119872minus1198981015840))

]]

]

)

minus1

)1198891198981015840

= 2120587int

119898

0

(1(21205961199031015840

ℎ

minus2120596211990310158403

ℎ+ 2120596 (119872 minus 119898

1015840)

radic1199031015840ℎ[120596211990310158403ℎ+ 4120596 (119872 minus 1198981015840)]

)

minus1

)1198891198981015840

= 2120587int

119898

0

((1

2+ 2120596(119872 minus 119898

1015840)2


2

minus1

2120596)


minus1

2120596)

minus1

)1198891198981015840

= 2120587[[

[

int

119898

0

1

4120596radic(119872 minus 1198981015840)2

minus 12120596

1198891198981015840

+ int

119898

0

(119872 minus 1198981015840)2

radic(119872 minus 1198981015840)2

minus 12120596

1198891198981015840

+int

119898

0

(119872 minus 1198981015840) 1198891198981015840]]

]


= 2120587[119872

2radic1198722 minus

1

2120596minus119872 minus 119898

1015840

2radic(119872 minus 1198981015840)

2

minus1

2120596

+1

2120596[ln119872+radic1198722 minus

1

2120596minus ln (119872 minus 119898)


2120596] +119872119898 minus

1198982

2]

(12)


1

That is

119868 = 21198681 (13)


119879119867=

21205961199032

+minus 1

8120587 (1205961199033++ 119903+)

=radic1198722 minus 12120596

2120587 (21198722 + 2119872radic1198722 minus 12120596 + 12120596)

(14)


(12)(120597119891(119903)120597119903)|119903=119903+


119878BH =119860

4+120587

120596ln 1198604 (15)



Δ119878 = 120587 [1199032

+(119872) minus 119903

2

+(119872 minus 119898)] = 119868 (16)


1198891199042= minus119891 (119903) 119889119905

2+ 119891(119903)

minus11198891199032 (17)


119903lowast= 119903 +

1

2120581+

ln119903 minus 119903+

119903+

+1

2120581minus


119903minus

(18)

in which 120581plusmn= (1198911015840(119903)2)|

119903=119903plusmn

= (21205961199032


plusmn(1205961199032

plusmn+1)) is the





119879 =1

2(119881 + 119880)

= 119890120581+119903(119903 minus 119903+

119903+

)

12


119903minus

)

120581+2120581minus

sinh 120581+119905

119877 =1

2(119881 minus 119880) = 119890

120581+119903(119903 minus 119903+

119903+

)

12


119903minus

)

120581+2120581minus

cosh 120581+119905

(19)


119889119903lowast= 119891(119903)

minus1119889119903 (20)


1198891199042= minus119889119905

2+ 1198891199032

lowast

= 120581minus2


119903+

)(119903minus


)

120581+120581minus

times [minus1198891198792+ 1198891198772]

(21)


119894119879 = 119890120581+119903(119903 minus 119903+

119903+

)

12


119903minus

)

120581+2120581minus

sin 120581+120591

119877 = 119890120581+119903(119903 minus 119903+

119903+

)

12


119903minus

)

120581+2120581minus

cos 120581+120591

(22)




119868 = int119901119894119889119902119894= intint

119901119894

0

1198891199011015840

119894119889119902119894= intint

119867

0

1198891198671015840

119902119894

119889119902119894

= intint

119867

0

1198891198671015840119889120591 + intint

119867

0

1198891198671015840

119903119889119903

(23)


dinate 119902119894with 119894 = 0 1 for 119902

0= 120591 and 119902


119903 = 119889119903119889120591 we have

119868 = 2intint

119867

0

1198891198671015840119889120591 (24)


119868 = 2120587int

119867

0

1198891198671015840

120581= ℎint

119867

0

1198891198671015840

119879BH= ℎ119878BH (25)



∮119901119889119902 = 2120587119899ℎ (26)


119878BH = 2120587119899 119899 = 1 2 3 (27)


Δ119878BH = 2120587 (28)


Δ119878BH =1

4Δ119860 +

120587

120596

4

119860Δ119860 (29)

we have

Δ119860 =Δ119878BH

14 + 4120587119860120596≃ 8120587(1 minus

4

1205961199032+

) (30)






Acknowledgments


References









































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




= 2120587[119872

2radic1198722 minus

1

2120596minus119872 minus 119898

1015840

2radic(119872 minus 1198981015840)

2

minus1

2120596

+1

2120596[ln119872+radic1198722 minus

1

2120596minus ln (119872 minus 119898)


2120596] +119872119898 minus

1198982

2]

(12)


1

That is

119868 = 21198681 (13)


119879119867=

21205961199032

+minus 1

8120587 (1205961199033++ 119903+)

=radic1198722 minus 12120596

2120587 (21198722 + 2119872radic1198722 minus 12120596 + 12120596)

(14)


(12)(120597119891(119903)120597119903)|119903=119903+


119878BH =119860

4+120587

120596ln 1198604 (15)



Δ119878 = 120587 [1199032

+(119872) minus 119903

2

+(119872 minus 119898)] = 119868 (16)


1198891199042= minus119891 (119903) 119889119905

2+ 119891(119903)

minus11198891199032 (17)


119903lowast= 119903 +

1

2120581+

ln119903 minus 119903+

119903+

+1

2120581minus


119903minus

(18)

in which 120581plusmn= (1198911015840(119903)2)|

119903=119903plusmn

= (21205961199032


plusmn(1205961199032

plusmn+1)) is the





119879 =1

2(119881 + 119880)

= 119890120581+119903(119903 minus 119903+

119903+

)

12


119903minus

)

120581+2120581minus

sinh 120581+119905

119877 =1

2(119881 minus 119880) = 119890

120581+119903(119903 minus 119903+

119903+

)

12


119903minus

)

120581+2120581minus

cosh 120581+119905

(19)


119889119903lowast= 119891(119903)

minus1119889119903 (20)


1198891199042= minus119889119905

2+ 1198891199032

lowast

= 120581minus2


119903+

)(119903minus


)

120581+120581minus

times [minus1198891198792+ 1198891198772]

(21)


119894119879 = 119890120581+119903(119903 minus 119903+

119903+

)

12


119903minus

)

120581+2120581minus

sin 120581+120591

119877 = 119890120581+119903(119903 minus 119903+

119903+

)

12


119903minus

)

120581+2120581minus

cos 120581+120591

(22)




119868 = int119901119894119889119902119894= intint

119901119894

0

1198891199011015840

119894119889119902119894= intint

119867

0

1198891198671015840

119902119894

119889119902119894

= intint

119867

0

1198891198671015840119889120591 + intint

119867

0

1198891198671015840

119903119889119903

(23)


dinate 119902119894with 119894 = 0 1 for 119902

0= 120591 and 119902


119903 = 119889119903119889120591 we have

119868 = 2intint

119867

0

1198891198671015840119889120591 (24)


119868 = 2120587int

119867

0

1198891198671015840

120581= ℎint

119867

0

1198891198671015840

119879BH= ℎ119878BH (25)



∮119901119889119902 = 2120587119899ℎ (26)


119878BH = 2120587119899 119899 = 1 2 3 (27)


Δ119878BH = 2120587 (28)


Δ119878BH =1

4Δ119860 +

120587

120596

4

119860Δ119860 (29)

we have

Δ119860 =Δ119878BH

14 + 4120587119860120596≃ 8120587(1 minus

4

1205961199032+

) (30)






Acknowledgments


References









































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





∮119901119889119902 = 2120587119899ℎ (26)


119878BH = 2120587119899 119899 = 1 2 3 (27)


Δ119878BH = 2120587 (28)


Δ119878BH =1

4Δ119860 +

120587

120596

4

119860Δ119860 (29)

we have

Δ119860 =Δ119878BH

14 + 4120587119860120596≃ 8120587(1 minus

4

1205961199032+

) (30)






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








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Documents

Research Article Entropy Spectrum of a KS Black Hole in IR ...downloads.hindawi.com/journals/ahep/2014/396453.pdf · Research Article Entropy Spectrum of a KS Black Hole in IR Modified