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Research ArticleCardy-Verlinde Formula and Its Self-GravitationalCorrections for Regular Black Holes
M Sharif and Rabia Saleem
Department of Mathematics University of the Punjab Quaid-e-Azam Campus Lahore 54590 Pakistan
Correspondence should be addressed to M Sharif msharifmathpuedupk
Received 28 March 2014 Accepted 3 May 2014 Published 21 May 2014
Academic Editor Ming Liu
Copyright copy 2014 M Sharif and R Saleem This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited The publication of this article was funded by SCOAP3
We check the consistency of the entropy of Bardeen and Ayon Beato-Garcıa-Bronnikov black holes with the entropy of particularconformal field theory via Cardy-Verlinde formula We also compute the first-order semiclassical corrections of this formula dueto self-gravitational effects by modifying pure extensive and Casimir energy in the context of Keski-Vakkuri Kraus and Wilczekanalysis It is concluded that the correction term remains positive for both black holes which leads to the violation of the holographicbound
1 Introduction
The idea of massive bodies from which nothing can escapedue to strong gravity was firstly proposed in 1783 by Michell[1] and named as ldquodark starsrdquo In 1967 Wheeler [2] usedthe term ldquoblack holerdquo for such objects Black hole (BH) isthe most important prediction of general relativity whichis detected by its effect on the nearby matter A BH canbe defined as a spacetime singularity which is covered byan event horizon Hawking [3] found that BHs also radiateparticles from their boundary with finite thermal spectrumdubbed after his name as Hawking radiations Several effortshave been made to visualize this spectrum by examining thequantum effects on a scalar particle [4] Taking quantumeffects into account BH evaporation converts pure quantumstates into mixed states known as ldquoinformation loss paradoxrdquo[5] Much work has been done to resolve this paradox usingfixed geometrical background during emission process
In 1995 Kraus and Wilczek [6 7] put forward ananalysis in which dynamical BH background was usedby treating the Hawking radiations as a tunneling pro-cess They found that Hawking radiations are modifiedand their spectrum becomes nonthermal after consideringself-gravitational interaction In this procedure total mass(Arnowitt-Deser-Misner (ADM)) remains constant whilethe mass of the Schwarzschild BH decreases due to emitted
radiations A specific Painleve coordinate transformation[8] is used in order to avoid the existence of singularityat the horizon Initially this analysis was implemented onspherically symmetric geometry (Schwarzschild type BHs119860(119903) = 119861
minus1
(119903)) to evaluate their corrected temperature andentropy An extra nonthermal term appears in thermody-namical quantities due to thismodificationThemodified BHtemperature also depends upon the emitted shell energy (120596)along with BH characteristics while entropy is different fromthe Bekenstein-Hawking entropy (119878) for the correspondingSchwarzschild BH
Vagenas [9] applied Keski-Vakkuri Kraus and Wilczek(KKW) analysis on non-Schwarzschild type BH using moregeneralized coordinate transformation This transformationallows studying the across horizonBHphysicswhose general-ization leads to exact expressions of thermodynamical quan-tities The temperature and entropy of non-Schwarzschildtype BH are not any more the Hawking temperature (119879
119867
)
and Bekenstein-Hawking entropy respectively Parikh andWilczek [10] were the first to utilize this technique to four-dimensional Schwarzschild BH Later this analysis was usedfor several BHs such as (119889 + 1)-dimensional anti-de Sitter(AdS) BH 2-dimensional AdS BH two-dimensional chargedand uncharged dilatonic BH (2 + 1)-dimensional chargedBTZBH spinning BTZBH andmagnetic stringy BH [11ndash17]
Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2014 Article ID 926589 7 pageshttpdxdoiorg1011552014926589
2 Advances in High Energy Physics
Holographic principle proposed by Susskind [18] is oneof the fundamental principles of quantum gravity It statesthat Bekenstein-Hawking entropy should be larger than theentropy associated with volume 119881(119878 le A4) where 119866 =
119888 = ℎ = 1 Generally radiations can be describedby using interacting conformal field theory (CFT) and theAdSCFT correspondence is the most common example ofthe holography Verlinde [19] proposed a universal formula(Cardy-Verlinde formula (CV)) which relates the entropy of acertain CFT (119878CFT) to its total energy (119864) and Casimir energy(119864119862
) valid for all dimensions The generalization of thisformula is the major outcome of AdSCFT correspondenceThis formula remains valid for topological dS Schwarzschild-dS Reissner-Nordstrom-dS Kerr-dS Kerr-Newman-dS BHsand so forth [20ndash26]
The issue of quantum corrections in entropy attractedmany researchers [27ndash29] Carlip [30] deduced the leadingorder quantum correction to the classical CV formula Setareand Vagenas [31 32] proved that this formula holds forAchucarro-Ortiz BH and calculated the self-gravitationalcorrections in 119878CFT in the framework of KKWanalysis Setareand Jamil [33] showed that the entropy of the charged rotatingBTZ BH can be expressed in terms of 119878CFT Darabi et al[34] found the self-gravitational corrections in this formulafor the charged BTZ BH Recently Abbas [35] showed thatthe entropy of noncommutative Schwarzschild BH can beexpressed in terms of CV formula
In this paper we study entropy of the Bardeen andAyon Beato-Garcıa-Bronnikov (ABGB) BHs in terms of CVformula and also find the first-order leading self-gravitationalcorrections The paper is organized as follows In Section 2we calculate quantities like mass potential temperature andentropy of both BHs We prove that the entropies of theseBHs coincide with that of CFT via CV formula Section 3 isdevoted to the computation of the self-gravitational correc-tions in the CFT entropy of these BHs using KKW analysisFinally we summarize the results in the last section
2 Entropy of Regular Black Holes viaCardy-Verlinde Formula
In this section we evaluate the entropy of the Bardeen andABGB BHs through the Cardy-Verlinde formula
21 Entropy of Bardeen Black Hole The four-dimensionalspherically symmetric BH is represented by the following lineelement
1198891199042
= minus1198601198891199052
+ 119860minus1
1198891199032
+ 1199032
1198891205792
+ 1199032sin21205791198891206012 (1)
where
119860 (119903) = 1 minus 2119872 (119903)
119903 (2)
For different choices of 119872(119903) this reduces to some well-known BHs Bardeen [36] proposed a model obeying weakenergy condition that can be interpreted as the solution of
magnetic monopole The mass function has the followingspecific choice for Bardeen BH
119872(119903) =1198981199033
(1199032 + 1198762)32
(3)
with mass119898 and charge 119876 For zero charges this mass func-tion leads to the Schwarzschild BHThe lapse function119860(119903) =0 provides a unique root (119903
+
= 2119872(119903+
)) corresponding tothe event horizon (outer horizon) Substituting this value of119872(119903) the event horizon takes the form
119903+
= 2119872(119903+
) = 21198981199033
+
(1199032+
+ 1198762)32
(4)
leading to the value of119898 as
119898 =
(1199032
+
+ 1198762
)32
21199032+
(5)
The corresponding electric potential (Φ+
) and the Hawkingtemperature can be calculated as
Φ+
=120597119872
120597119876
10038161003816100381610038161003816100381610038161003816119903=119903
+
= minus3
2(
119876119903+
1199032+
+ 1198762)
119879119867
=1
4120587
119889119860
119889119903
10038161003816100381610038161003816100381610038161003816119903=119903+
=1
4120587119903+
(1199032
+
minus 21198762
1199032+
+ 1198762)
(6)
The corresponding area and entropy are given as
A = intradic119892120579120579
119892120601120601
119889120579 119889120601 = 41205871199032
+
119878 =A
4= 1205871199032
+
= 120587(21198981199033
+
(1199032+
+ 1198762)32
)
2
(7)
The generalized Cardy-Verlinde formula is defined as [19]
119878CFT =2120587119877
radic119886119887
radic119864119862
(2119864 minus 119864119862
) (8)
where 119886 and 119887 are arbitrary positive constants and 119877 isthe radius of the sphere The Casimir energy is generatedby quantum fluctuations of CFT at finite volume while itdisappears when volume becomes infinite Casimir effectsare usually significant at zero temperatures but can also bediscussed at its finite values of the temperature Thus anextensive (in thermodynamical system the energy 119864(119878 119881)is called extensive when it satisfies the relation 119864(120582119878 120582119881) =120582119864(119878 119881)) part is added in the total energy expressed as
119864 (119878 119881) = 119864119864
(119878 119881) +1
2119864119862
(119878 119881) (9)
where 119864119864
is the pure extensive part of energy and the factor12 is used for convenience [19]
The violation of Euler identity is given as
119864 = 119881(120597119864
120597119881)
119878
+ 119878(120597119864
120597119878)
119881
(10)
Advances in High Energy Physics 3
which provides 119899-dimensional Casimir energy in the follow-ing form
119864119862
= 119899 (119864 + 119875119881 minus 119879119878 minus Φ+
119876) (11)where pressure is defined as119875 = 119864119899119881The product of energyand radius (ER) is independent of volume due to conformalinvariance which is satisfied by both 119864
119864
and 119864119862
Hence bothenergies can be expressed in terms of 119877 and 119878 in arbitrarydimensions as follows [19]
119864119862
=119887
21205871198771198781minus1119899
119864119864
=119886
41205871198771198781+1119899
(12)
Taking 119899 = 1 and using the above expressions of 119875 119879119867
119878 Φ+
in (11) the Casimir energy takes the form
119864119862
= 2119864 minus119903+
4(1199032
+
minus 21198762
1199032+
+ 1198762) + (
31198762
119903+
1199032+
+ 1198762) (13)
Using mass energy relation (119864 = 119872) this reduces to
119864119862
= 2119872 minus119903+
4(1199032
+
minus 21198762
1199032+
+ 1198762) + (
31198762
119903+
1199032+
+ 1198762) (14)
The extensive energy is found through (9) and (13) as
2119864119864
= 2119864 minus 119864119862
= minus119903+
4(1199032
+
minus 21198762
1199032+
+ 1198762) + (
31198762
119903+
1199032+
+ 1198762) (15)
Using (7) the generalized formulas of Casimir and extensiveenergy in one dimension can be written as
119864119862
=119887
2120587119877 119864
119864
=120587119886
41198771199034
+
(16)
Comparing the Casimir as well as extensive energy given in(14)ndash(16) we obtain two different spherical radii as
119877 = 1205871198861199034
+
[
(1199032
+
+ 1198762
)
2120587119903+
(1199032+
minus 21198762) minus 31198762119903+
]
119877 =119887
120587
times [
(1199032
+
+ 1198762
)
4119872 (1199032+
+ 1198762) minus 2119903+
(1199032+
minus 21198762) minus 31198762119903+
]
(17)
Multiplication of the above two equations provides119877
= (119886119887119903+
(1199032
+
+ 1198762
)2
times ((2120587119903+
(1199032
+
minus 21198762
) minus 31198762
119903+
)
times (4119872(1199032
+
+ 1198762
) minus 2119903+
(1199032
+
minus 21198762
) minus 31198762
119903+
))minus1
)
12
(18)Substituting (14) (15) and (18) in CV formula the final resultis
119878CFT = 1205871199032
+
= 119878 (19)This shows that the entropy of Bardeen BH can be expressedin terms of CV formula in the context of CFT
22 Entropy of Ayon Beato-Garcıa-Bronnikov Black HoleAyon Beato and Garcıa [37] and Bronnikov [38] proposed anonsingular BH by solving the system of equations coupledwith nonlinear electrodynamics and gravity For this BH themass function is given as
119872(119903) = 119898[1 minus tanh( 1198762
2119898119903)] (20)
which also reduces to the Schwarzschild BH for 119876 = 0 Theevent horizon has the following form
119903+
= 2119872(119903+
) = 2119898[1 minus tanh( 1198762
2119898119903+
)] (21)
The corresponding electric potential Hawking temperatureand entropy turn out to be
Φ+
=120597119872
120597119876
10038161003816100381610038161003816100381610038161003816119903=119903
+
= minus119876
119903+
sec ℎ2 ( 1198762
2119898119903+
)
119879119867
=1
4120587
119889119860
119889119903
10038161003816100381610038161003816100381610038161003816119903=119903+
=1
41205871199032+
(119903+
+ 119876Φ+
)
119878 =A
4= 1205871199032
+
= 120587(2119898[1 minus tanh( 1198762
2119898119903+
)])
2
(22)
Using (11) the Casimir energy can be written as
119864119862
= 2119864 minus1
4119903+
+5
4(1198762
119903+
sec ℎ2 ( 1198762
2119898119903+
)) (23)
For 119864 = 119872 this reduces to
119864119862
=3
2119872 +
5
4(1198762
119903+
sec ℎ2 ( 1198762
2119898119903+
)) (24)
The extensive energy is given as
2119864119864
= 2119864 minus 119864119862
=1
4119903+
minus5
4(1198762
119903+
sec ℎ2 ( 1198762
2119898119903+
)) (25)
Comparison of (24) and (25) with (16) leads to
119877 =120587119886
41199034
+
[1
(18) 119903+
minus (58) ((1198762119903+
) sec ℎ2 (11987622119898119903+
))]
119877 =119887
2120587[
1
(32)119872 + (54) ((1198762119903+
) sec ℎ2 (11987622119898119903+
))]
(26)
yielding the unique radius
119877 =radic119886119887
2radic21199032
+
times [1 times ((3
2119872 +
5
4(1198762
119903+
sec ℎ2 ( 1198762
2119898119903+
)))
times(1
8119903+
minus5
8(1198762
119903+
sec ℎ2 ( 1198762
2119898119903+
))))
minus12
]
(27)
4 Advances in High Energy Physics
Inserting (24)ndash(27) in (8) we find that the entropy of CFT isequivalent to the entropy of ABGB BH as
119878CFT = 1205871199032
+
= 119878 (28)
3 Self-Gravitational Corrections tothe Cardy-Verlinde Formula
In this section we evaluate the self-gravitational correctionsin the CV formula for both BHs using KKW analysis Thisleads to modifying all the quantities in the above sectionunder self-gravitational effects
31 Self-Gravitational Corrections for Bardeen Black HoleHere we evaluate the self-gravitational corrections in CVformula for Bardeen BH The modified Casimir energy (119864
119862
)
is defined as
119864119862
= 2119864 minus 119879bh119878bh minus 119876bhΦbh (29)
where 119879bh and 119878bh stand for modified temperature andentropy respectively The third term of (11) is modified to119879bh119878bh whereas the total energy charge radius of the sphereand electric potential of BH remain invariant under thiseffect In order to evaluate the corrections in BH temperatureand entropy we use the KKW analysis In this analysis thetotalmass is kept fixed while BHmass is assumed to fluctuatebecause a shell of energy 120596 radiates massless particles andhence BH mass reduces to 119872 minus 120596 This shell energy doesnot contain charge due to which electric potential remainsthe same that is 119876bhΦbh = 119876Φ+ Applying this analysis weobtain a relationship between BH entropy and Bekenstein-Hawking entropy as
119878bh = 119878 minus 41205871198722
[1 minus (1 minus120596
119872)
2
] (30)
The corresponding self-gravitational corrected temperature isgiven by
119879bh =120596
41205871198722[1 minus (1 minus
120596
119872)
2
]
minus1
(31)
which reduces to the Hawking temperature by consideringcorrections up to first-order in 120596
Thus the self-gravitational corrections in 119879bh119878bh take theform
119879bh119878bh = 119879119867119878 minus2
119903+
(1199032
+
minus 21198762
1199032+
+ 1198762)119872120596 (32)
Substituting this value in (29) the modified Casimir energyturns out to be
119864119862
= 2119872 minus119903+
4(1199032
+
minus 21198762
1199032+
+ 1198762) +
3
2(119903+
1198762
1199032+
+ 1198762)
minus (1199032
+
minus 21198762
1199032+
+ 1198762)119872120596
= 119864119862
+ (1199032
+
minus 21198762
1199032+
+ 1198762)119872120596
(33)
where the second term is the correction term The modifica-tion in extensive part of the energy can be obtained by addingthe term 2119864 in the above expression as
119864119864
=119903+
8(1199032
+
minus 21198762
1199032+
+ 1198762) minus
3
4(119903+
1198762
1199032+
+ 1198762) minus (
1199032
+
minus 21198762
1199032+
+ 1198762)119872120596
= 119864119864
minus (1199032
+
minus 21198762
1199032+
+ 1198762)119872120596
(34)
The extensive energy (16) can also be modified in the contextof KKW analysis as follows
119864119864
=119886
41205871198771198782
bh =120587119886
41198771199034
out =120587119886
41198771199034
+
(1 minus120596
119872) (35)
Inserting (18) (33) and (34) in modified CV formula that is
119878CFT =2120587119877
radic119886119887
radic119864119862
(2119864 minus 119864119862
) (36)
it follows that
119878CFT = 119878(((8119872119903+ (1199032
+
+ 1198762
) minus 1199032
+
(1199032
+
minus 21198762
)
+64
119903+
(1199032
+
+ 1198762
)1198722
120596 + 8119903+
(1199032
+
minus 21198762
)119872120596
minus8 (1199032
+
minus 21198762
)119872120596) + 121198762
1199032
+
+ 361198764
1199032
+
)
times [
[
(1199032
+
minus 21198762
)minus2
(4119872 (1199032+
+ 1198762) minus 120572) 120572
]
]
)
12
(37)
where 120572 = (2120587119903+
(1199032
+
minus 21198762
) + 31198762
119903+
) This gives the correc-tions in CV formula due to the effects of self-gravitation
The total energy is the combination of extensive andCasimir energies so 119864 gt 119864
119862
which implies that the term2119864 minus 119864
119862
is positive As the correction term is positive so themodified term 119864
119862
(2119864 minus 119864119862
) is greater in magnitude than theoriginal one The self-gravitational correction of CV formulafor Bardeen BH remains positive as the term inside squareroot is real and positive The above term can be expanded upto the first-order in 120596 as follows
119864119862
(2119864 minus 119864119862
) = 119864119862
(2119864 minus 119864119862
)
minus (119864119862
minus (2119864 minus 119864119862
)) (1199032
+
minus 21198762
1199032+
+ 1198762)119872120596
(38)
which relates the modified self-gravitational energy with theoriginal energy It is seen that the self-gravitational modifiedtemperature and entropy derived in the context of KKWanalysis of Bardeen BH are different from the Bekenstein-Hawking temperature and entropy respectively The BHentropy described by CV formula violates the holographicbound as 119878CFT gt 119878 gt 119878bh [39 40]
Advances in High Energy Physics 5
32 Self-Gravitational Corrections for Ayon Beato-Garcıa-Bronnikov Black Hole Now we consider ABGB BH tocalculate the self-gravitational corrections in the CV formulausing KKW analysis The corresponding self-gravitationalcorrections in 119879bh119878bh are
119879bh119878bh = 119879119867119878 minus120596
2119872(119903+
+ 119876Φ+
) (39)
Equation (29) gives the modified Casimir energy as
119864119862
=3
2119872 +
5
4(1198762
119903+
sec ℎ2 ( 1198762
2119898119903+
)) +2
1199032+
(119903+
+ 119876Φ+
)119872120596
= 119864119862
+2
1199032+
(119903+
+ 119876Φ+
)119872120596
(40)
where the second term is the correction term Additionallythe modified extensive energy can be written as
2119864 minus 119864119862
= 2119864 minus 119864119862
minus2
1199032+
(119903+
+ 119876Φ+
)119872120596
119864119864
= 119864119864
minus1
1199032+
(119903+
+ 119876Φ+
)119872120596
(41)
Inserting (27) (40) and (41) in modified CV formula weobtain the self-gravitational corrected CV formula for ABGBBH as follows
119878CFT
= 119878[
[
((3
8119872119903+
minus4
1199032+
(119903+
+ 119876Φ+
)1198722
120596
minus25
16
1198764
1199032+
sec ℎ4 ( 1198762
2119898119903+
))
times ((1
8119903+
minus5
8
1198764
1199032+
sec ℎ4 ( 1198762
2119898119903+
))
times (3
2119872 +
5
4
1198764
1199032+
sec ℎ4 ( 1198762
2119898119903+
)))
minus1
)
12
]
]
(42)
As the correction term (21199032
+
)(119903+
+119876Φ+
) gt 0 this implies that119864119862
(2119864 minus 119864119862
) gt 119864119862
(2119864 minus 119864119862
) The corresponding first-orderexpansion of the correction term is
119864119862
(2119864 minus 119864119862
) = 119864119862
(2119864 minus 119864119862
)
+2
1199032+
(119903+
+ 119876Φ+
) (2119864 minus 119864119862
)119872120596
(43)
This shows that the ABGB BH entropy described by CVformula also violates the holographic bound
4 Concluding Remarks
The description of BH entropy by CV formula and itsquantum corrections have attained much attention Therehave been a number of papers to find the semiclassicalcorrections in this formula for different BHs using varioustechniques such as self-gravitational effects [41] generalizeduncertainty principle [42] space noncommutativity [43] andquantum corrections [44] The self-gravitational correctionallows one to study across horizon physics and provides thesolution of information loss paradox
This paper is devoted to study this analysis forthe Bardeen and ABGB BHs which correspond to theSchwarzschild BH for zero charges Firstly we have evaluatedthe thermodynamical quantities such as electric potentialHawking temperature and Bekenstein-Hawking entropy ofboth of the BHs It is found that there exists unique eventhorizon whose entropy can be expressed in the form of CVformula indicating that this formula holds on both eventhorizons
Secondly we have used the self-gravitational effect toevaluate the temperature and entropy of these BHs inthe context of KKW analysis The corrections in the CVformula due to self-gravitational effects are evaluated bytaking dynamical background of the BH The correctionsare restricted up to the linear order in 120596 for the zerothorder the modified temperature corresponds to the Hawkingtemperature We have also found the corrections in all thequantities which can be modified under self-gravitationaleffect that is pure extensive energy and the Casimir energyThe electric potential charge spherical radius and totalenergy remain invariant under this effect It turns out that theself-gravitational correction term in modified CV formula ispositive for both BHs as the modified term is greater than theoriginal It is interesting tomention here that the positive self-gravitational corrections for regular BHs do not satisfy theinequality 119878CFT gt 119878 gt 119878bh which proves that the holographicbound is not universal
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the Higher EducationCommission Islamabad Pakistan for its financial supportthrough the Indigenous PhD Fellowship for 5K ScholarsPhase-II Batch-I
References
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6 Advances in High Energy Physics
for that purposerdquoPhilosophical Transactions of the Royal Societyvol 74 pp 35ndash57 1784
[2] J AWheeler ldquoOur universe the known and the unknownrdquoTheAmerican Scientist vol 56 p 1 1968
[3] S W Hawking ldquoBlack hole explosionsrdquo Nature vol 248 no5443 pp 30ndash31 1974
[4] J B Hartle and S W Hawking ldquoPath-integral derivation ofblack-hole radiancerdquo Physical Review D vol 13 no 8 pp 2188ndash2203 1976
[5] Q Q Jiang and X Cai ldquoRemarks on self-interaction correctionto black hole radiationrdquo Journal of High Energy Physics vol2003 no 11 article 110 2009
[6] P Kraus and F Wilczek ldquoSelf-interaction correction to blackhole radiancerdquo Nuclear Physics B vol 433 no 2 pp 403ndash4201995
[7] P Kraus and F Wilczek ldquoEffect of self-interaction on chargedblack hole radiancerdquo Nuclear Physics B vol 437 no 1 pp 231ndash242 1995
[8] P Painleve ldquoLamecanique classique et la theorie de la relativiterdquoComptes Rendus de lrsquoAcademie des Sciences vol 173 pp 677ndash680 1921
[9] E C Vagenas ldquoGeneralization of the KKW analysis for blackhole radiationrdquo Physics Letters B vol 559 no 1-2 pp 65ndash732003
[10] M K Parikh and F Wilczek ldquoHawking radiation as tunnelingrdquoPhysical Review Letters vol 85 no 24 pp 5042ndash5045 2000
[11] S Hemming and E Keski-Vakkuri ldquoHawking radiation fromAdS black holesrdquo Physical Review D vol 64 no 4 Article ID044006 8 pages 2001
[12] Y Kwon ldquoHawking radiation in AdS2
black holerdquo Il NuovoCimento della Societa Italiana di Fisica B Serie 12 vol 115 no 4pp 469ndash471 2000
[13] E C Vagenas ldquoAre extremal 2D black holes really frozenrdquoPhysics Letters B vol 503 no 3-4 pp 399ndash403 2001
[14] E C Vagenas ldquoTwo-dimensional dilatonic black holes andHawking radiationrdquo Modern Physics Letters A Particles andFields Gravitation Cosmology Nuclear Physics vol 17 no 10pp 609ndash618 2002
[15] E C Vagenas ldquoSemiclassical corrections to the Bekenstein-Hawking entropy of the BTZ black hole via self-gravitationrdquoPhysics Letters B vol 533 no 3-4 pp 302ndash306 2002
[16] A J M Medved ldquoRadiation via tunnelling in the charged BTZblack holerdquo Classical and Quantum Gravity vol 19 no 3 pp589ndash598 2002
[17] A J M Medved ldquoRadiation via tunneling from a de Sittercosmological horizonrdquo Physical ReviewD vol 66 no 12 ArticleID 124009 7 pages 2002
[18] L Susskind ldquoTheworld as a hologramrdquo Journal ofMathematicalPhysics vol 36 no 11 pp 6377ndash6396 1995
[19] E Verlinde ldquoOn the holographic principle in a radiationdominateduniverserdquo httparxivorgabshepth0008140
[20] D Birmingham and S Mokhtari ldquoThe Cardy-Verlinde formulaand Taub-bolt-AdS spacetimesrdquo Physics Letters B vol 508 no3-4 pp 365ndash368 2001
[21] D Klemm A C Petkou and G Siopsis ldquoEntropy boundsmonotonicity properties and scaling in CFTsrdquo Nuclear PhysicsB vol 601 no 1-2 pp 380ndash394 2001
[22] B Wang E Abdalla and R-K Su ldquoRelating Friedmannequation to Cardy formula in universes with cosmologicalconstantrdquo Physics Letters B vol 503 no 3-4 pp 394ndash398 2001
[23] M R Setare ldquoThe Cardy-Verlinde formula and entropy oftopological Reissner-Nordstrom black holes in de Sitter spacesrdquoModern Physics Letters A Particles and Fields GravitationCosmology Nuclear Physics vol 17 no 32 pp 2089ndash2094 2002
[24] M R Setare andM B Altaie ldquoTheCardy-Verlinde formula andentropy of topological Kerr-Newman black holes in de Sitterspacesrdquo The European Physical Journal C Particles and Fieldsvol 30 no 2 pp 273ndash277 2003
[25] M R Setare and R Mansouri ldquoHolographic thermodynamicson the brane in topological Reissner-Nordstromde Sitter spacerdquoInternational Journal of Modern Physics A vol 18 no 24 pp4443ndash4450 2003
[26] C O Lee ldquoCardy-Verlinde formula in Taub-NUTbolt-(A)dSspacerdquo Physics Letters B vol 670 no 2 pp 146ndash149 2008
[27] A J M Medved ldquoQuantum-corrected Cardy entropy forgeneric (1 + 1)-dimensional gravityrdquo Classical and QuantumGravity vol 19 no 9 pp 2503ndash2513 2002
[28] SMukherji and S S Pal ldquoLogarithmic corrections to black holeentropy and AdSCFT correspondencerdquo Journal of High EnergyPhysics vol 2002 no 5 article 026 2002
[29] J E Lidsey S Nojiri S D Odintsov and S OgushildquoThe AdSCFT correspondence and logarithmic correctionsto braneworld cosmology and the Cardy-Verlinde formulardquoPhysics Letters B vol 544 no 3-4 pp 337ndash345 2002
[30] S Carlip ldquoLogarithmic corrections to black hole entropy fromthe Cardy formulardquo Classical and Quantum Gravity vol 17 no20 pp 4175ndash4186 2000
[31] M R Setare and E C Vagenas ldquoCardy-Verlinde formula andAchucarro-Ortiz black holerdquo Physical Review D vol 68 no 6Article ID 064014 5 pages 2003
[32] MR Setare andECVagenas ldquoSelf-gravitational corrections tothe Cardy-Verlinde formula of the Achucarro-Ortiz black holerdquoPhysics Letters B vol 584 no 1-2 pp 127ndash132 2004
[33] M R Setare and M Jamil ldquoThe Cardy-Verlinde formula andentropy of the charged rotating BTZ black holerdquo Physics LettersB vol 681 no 5 pp 469ndash471 2009
[34] F Darabi M Jamil and M R Setare ldquoSelf-gravitationalcorrections to theCardy-Verlinde formula of chargedBTZblackholerdquoModern Physics Letters A Particles and Fields GravitationCosmology Nuclear Physics vol 26 no 14 pp 1047ndash1057 2011
[35] G Abbas ldquoCardy-Verlinde formula of noncommutativeSchwarzschild black holerdquo Advances in High Energy Physicsvol 2014 Article ID 306256 4 pages 2014
[36] J Bardeen ldquoNon-singular general relativistic gravitational col-lapserdquo in Proceedings of the 5th International Conference onGravitation and the Theory of Relativity Tbilisi Unversty PressTbilisi Georgia September 1968
[37] E Ayon-Beato and A Garcıa ldquoNew regular black hole solutionfrom nonlinear electrodynamicsrdquo Physics Letters B vol 464 no1-2 pp 25ndash29 1999
[38] K A Bronnikov ldquoComment on lsquoregular black hole in gen-eral relativity coupled to nonlinear electrodynamicsrsquordquo PhysicalReview Letters vol 85 p 4641 2000
[39] R Bousso ldquoA covariant entropy conjecturerdquo Journal of HighEnergy Physics vol 1999 no 7 article 004 1999
[40] G rsquot Hooft ldquoDimensional reduction in quantumgravityrdquohttparxivorgabsgr-qc9310026
[41] M R Setare and E C Vagenas ldquoSelf-gravitational correctionsto the Cardy-Verlinde formula and the FRW brane cosmologyin SdS
5
bulkrdquo International Journal of Modern Physics A vol 20no 30 pp 7219ndash7232 2005
Advances in High Energy Physics 7
[42] M R Setare ldquoThe generalized uncertainty principle and cor-rections to the Cardy-Verlinde formula in SAdS
5
black holesrdquoInternational Journal of Modern Physics A vol 21 no 6 p 13252006
[43] M R Setare ldquoSpace noncommutativity corrections to theCardy-Verlinde formulardquo International Journal of ModernPhysics A vol 21 no 13-14 p 3007 2006
[44] M Sharif and W Javed ldquoQuantum corrections for ABGB blackholerdquoAstrophysics and Space Science vol 337 no 1 pp 335ndash3412012
Submit your manuscripts athttpwwwhindawicom
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ThermodynamicsJournal of
![Page 2: Research Article Cardy-Verlinde Formula and Its Self ...downloads.hindawi.com/journals/ahep/2014/926589.pdf · e generalized Cardy-Verlinde formula is de ned as [ ] CFT = 2 % & '*](https://reader035.vdocuments.net/reader035/viewer/2022071219/6056d064e08cc116434a5561/html5/thumbnails/2.jpg)
2 Advances in High Energy Physics
Holographic principle proposed by Susskind [18] is oneof the fundamental principles of quantum gravity It statesthat Bekenstein-Hawking entropy should be larger than theentropy associated with volume 119881(119878 le A4) where 119866 =
119888 = ℎ = 1 Generally radiations can be describedby using interacting conformal field theory (CFT) and theAdSCFT correspondence is the most common example ofthe holography Verlinde [19] proposed a universal formula(Cardy-Verlinde formula (CV)) which relates the entropy of acertain CFT (119878CFT) to its total energy (119864) and Casimir energy(119864119862
) valid for all dimensions The generalization of thisformula is the major outcome of AdSCFT correspondenceThis formula remains valid for topological dS Schwarzschild-dS Reissner-Nordstrom-dS Kerr-dS Kerr-Newman-dS BHsand so forth [20ndash26]
The issue of quantum corrections in entropy attractedmany researchers [27ndash29] Carlip [30] deduced the leadingorder quantum correction to the classical CV formula Setareand Vagenas [31 32] proved that this formula holds forAchucarro-Ortiz BH and calculated the self-gravitationalcorrections in 119878CFT in the framework of KKWanalysis Setareand Jamil [33] showed that the entropy of the charged rotatingBTZ BH can be expressed in terms of 119878CFT Darabi et al[34] found the self-gravitational corrections in this formulafor the charged BTZ BH Recently Abbas [35] showed thatthe entropy of noncommutative Schwarzschild BH can beexpressed in terms of CV formula
In this paper we study entropy of the Bardeen andAyon Beato-Garcıa-Bronnikov (ABGB) BHs in terms of CVformula and also find the first-order leading self-gravitationalcorrections The paper is organized as follows In Section 2we calculate quantities like mass potential temperature andentropy of both BHs We prove that the entropies of theseBHs coincide with that of CFT via CV formula Section 3 isdevoted to the computation of the self-gravitational correc-tions in the CFT entropy of these BHs using KKW analysisFinally we summarize the results in the last section
2 Entropy of Regular Black Holes viaCardy-Verlinde Formula
In this section we evaluate the entropy of the Bardeen andABGB BHs through the Cardy-Verlinde formula
21 Entropy of Bardeen Black Hole The four-dimensionalspherically symmetric BH is represented by the following lineelement
1198891199042
= minus1198601198891199052
+ 119860minus1
1198891199032
+ 1199032
1198891205792
+ 1199032sin21205791198891206012 (1)
where
119860 (119903) = 1 minus 2119872 (119903)
119903 (2)
For different choices of 119872(119903) this reduces to some well-known BHs Bardeen [36] proposed a model obeying weakenergy condition that can be interpreted as the solution of
magnetic monopole The mass function has the followingspecific choice for Bardeen BH
119872(119903) =1198981199033
(1199032 + 1198762)32
(3)
with mass119898 and charge 119876 For zero charges this mass func-tion leads to the Schwarzschild BHThe lapse function119860(119903) =0 provides a unique root (119903
+
= 2119872(119903+
)) corresponding tothe event horizon (outer horizon) Substituting this value of119872(119903) the event horizon takes the form
119903+
= 2119872(119903+
) = 21198981199033
+
(1199032+
+ 1198762)32
(4)
leading to the value of119898 as
119898 =
(1199032
+
+ 1198762
)32
21199032+
(5)
The corresponding electric potential (Φ+
) and the Hawkingtemperature can be calculated as
Φ+
=120597119872
120597119876
10038161003816100381610038161003816100381610038161003816119903=119903
+
= minus3
2(
119876119903+
1199032+
+ 1198762)
119879119867
=1
4120587
119889119860
119889119903
10038161003816100381610038161003816100381610038161003816119903=119903+
=1
4120587119903+
(1199032
+
minus 21198762
1199032+
+ 1198762)
(6)
The corresponding area and entropy are given as
A = intradic119892120579120579
119892120601120601
119889120579 119889120601 = 41205871199032
+
119878 =A
4= 1205871199032
+
= 120587(21198981199033
+
(1199032+
+ 1198762)32
)
2
(7)
The generalized Cardy-Verlinde formula is defined as [19]
119878CFT =2120587119877
radic119886119887
radic119864119862
(2119864 minus 119864119862
) (8)
where 119886 and 119887 are arbitrary positive constants and 119877 isthe radius of the sphere The Casimir energy is generatedby quantum fluctuations of CFT at finite volume while itdisappears when volume becomes infinite Casimir effectsare usually significant at zero temperatures but can also bediscussed at its finite values of the temperature Thus anextensive (in thermodynamical system the energy 119864(119878 119881)is called extensive when it satisfies the relation 119864(120582119878 120582119881) =120582119864(119878 119881)) part is added in the total energy expressed as
119864 (119878 119881) = 119864119864
(119878 119881) +1
2119864119862
(119878 119881) (9)
where 119864119864
is the pure extensive part of energy and the factor12 is used for convenience [19]
The violation of Euler identity is given as
119864 = 119881(120597119864
120597119881)
119878
+ 119878(120597119864
120597119878)
119881
(10)
Advances in High Energy Physics 3
which provides 119899-dimensional Casimir energy in the follow-ing form
119864119862
= 119899 (119864 + 119875119881 minus 119879119878 minus Φ+
119876) (11)where pressure is defined as119875 = 119864119899119881The product of energyand radius (ER) is independent of volume due to conformalinvariance which is satisfied by both 119864
119864
and 119864119862
Hence bothenergies can be expressed in terms of 119877 and 119878 in arbitrarydimensions as follows [19]
119864119862
=119887
21205871198771198781minus1119899
119864119864
=119886
41205871198771198781+1119899
(12)
Taking 119899 = 1 and using the above expressions of 119875 119879119867
119878 Φ+
in (11) the Casimir energy takes the form
119864119862
= 2119864 minus119903+
4(1199032
+
minus 21198762
1199032+
+ 1198762) + (
31198762
119903+
1199032+
+ 1198762) (13)
Using mass energy relation (119864 = 119872) this reduces to
119864119862
= 2119872 minus119903+
4(1199032
+
minus 21198762
1199032+
+ 1198762) + (
31198762
119903+
1199032+
+ 1198762) (14)
The extensive energy is found through (9) and (13) as
2119864119864
= 2119864 minus 119864119862
= minus119903+
4(1199032
+
minus 21198762
1199032+
+ 1198762) + (
31198762
119903+
1199032+
+ 1198762) (15)
Using (7) the generalized formulas of Casimir and extensiveenergy in one dimension can be written as
119864119862
=119887
2120587119877 119864
119864
=120587119886
41198771199034
+
(16)
Comparing the Casimir as well as extensive energy given in(14)ndash(16) we obtain two different spherical radii as
119877 = 1205871198861199034
+
[
(1199032
+
+ 1198762
)
2120587119903+
(1199032+
minus 21198762) minus 31198762119903+
]
119877 =119887
120587
times [
(1199032
+
+ 1198762
)
4119872 (1199032+
+ 1198762) minus 2119903+
(1199032+
minus 21198762) minus 31198762119903+
]
(17)
Multiplication of the above two equations provides119877
= (119886119887119903+
(1199032
+
+ 1198762
)2
times ((2120587119903+
(1199032
+
minus 21198762
) minus 31198762
119903+
)
times (4119872(1199032
+
+ 1198762
) minus 2119903+
(1199032
+
minus 21198762
) minus 31198762
119903+
))minus1
)
12
(18)Substituting (14) (15) and (18) in CV formula the final resultis
119878CFT = 1205871199032
+
= 119878 (19)This shows that the entropy of Bardeen BH can be expressedin terms of CV formula in the context of CFT
22 Entropy of Ayon Beato-Garcıa-Bronnikov Black HoleAyon Beato and Garcıa [37] and Bronnikov [38] proposed anonsingular BH by solving the system of equations coupledwith nonlinear electrodynamics and gravity For this BH themass function is given as
119872(119903) = 119898[1 minus tanh( 1198762
2119898119903)] (20)
which also reduces to the Schwarzschild BH for 119876 = 0 Theevent horizon has the following form
119903+
= 2119872(119903+
) = 2119898[1 minus tanh( 1198762
2119898119903+
)] (21)
The corresponding electric potential Hawking temperatureand entropy turn out to be
Φ+
=120597119872
120597119876
10038161003816100381610038161003816100381610038161003816119903=119903
+
= minus119876
119903+
sec ℎ2 ( 1198762
2119898119903+
)
119879119867
=1
4120587
119889119860
119889119903
10038161003816100381610038161003816100381610038161003816119903=119903+
=1
41205871199032+
(119903+
+ 119876Φ+
)
119878 =A
4= 1205871199032
+
= 120587(2119898[1 minus tanh( 1198762
2119898119903+
)])
2
(22)
Using (11) the Casimir energy can be written as
119864119862
= 2119864 minus1
4119903+
+5
4(1198762
119903+
sec ℎ2 ( 1198762
2119898119903+
)) (23)
For 119864 = 119872 this reduces to
119864119862
=3
2119872 +
5
4(1198762
119903+
sec ℎ2 ( 1198762
2119898119903+
)) (24)
The extensive energy is given as
2119864119864
= 2119864 minus 119864119862
=1
4119903+
minus5
4(1198762
119903+
sec ℎ2 ( 1198762
2119898119903+
)) (25)
Comparison of (24) and (25) with (16) leads to
119877 =120587119886
41199034
+
[1
(18) 119903+
minus (58) ((1198762119903+
) sec ℎ2 (11987622119898119903+
))]
119877 =119887
2120587[
1
(32)119872 + (54) ((1198762119903+
) sec ℎ2 (11987622119898119903+
))]
(26)
yielding the unique radius
119877 =radic119886119887
2radic21199032
+
times [1 times ((3
2119872 +
5
4(1198762
119903+
sec ℎ2 ( 1198762
2119898119903+
)))
times(1
8119903+
minus5
8(1198762
119903+
sec ℎ2 ( 1198762
2119898119903+
))))
minus12
]
(27)
4 Advances in High Energy Physics
Inserting (24)ndash(27) in (8) we find that the entropy of CFT isequivalent to the entropy of ABGB BH as
119878CFT = 1205871199032
+
= 119878 (28)
3 Self-Gravitational Corrections tothe Cardy-Verlinde Formula
In this section we evaluate the self-gravitational correctionsin the CV formula for both BHs using KKW analysis Thisleads to modifying all the quantities in the above sectionunder self-gravitational effects
31 Self-Gravitational Corrections for Bardeen Black HoleHere we evaluate the self-gravitational corrections in CVformula for Bardeen BH The modified Casimir energy (119864
119862
)
is defined as
119864119862
= 2119864 minus 119879bh119878bh minus 119876bhΦbh (29)
where 119879bh and 119878bh stand for modified temperature andentropy respectively The third term of (11) is modified to119879bh119878bh whereas the total energy charge radius of the sphereand electric potential of BH remain invariant under thiseffect In order to evaluate the corrections in BH temperatureand entropy we use the KKW analysis In this analysis thetotalmass is kept fixed while BHmass is assumed to fluctuatebecause a shell of energy 120596 radiates massless particles andhence BH mass reduces to 119872 minus 120596 This shell energy doesnot contain charge due to which electric potential remainsthe same that is 119876bhΦbh = 119876Φ+ Applying this analysis weobtain a relationship between BH entropy and Bekenstein-Hawking entropy as
119878bh = 119878 minus 41205871198722
[1 minus (1 minus120596
119872)
2
] (30)
The corresponding self-gravitational corrected temperature isgiven by
119879bh =120596
41205871198722[1 minus (1 minus
120596
119872)
2
]
minus1
(31)
which reduces to the Hawking temperature by consideringcorrections up to first-order in 120596
Thus the self-gravitational corrections in 119879bh119878bh take theform
119879bh119878bh = 119879119867119878 minus2
119903+
(1199032
+
minus 21198762
1199032+
+ 1198762)119872120596 (32)
Substituting this value in (29) the modified Casimir energyturns out to be
119864119862
= 2119872 minus119903+
4(1199032
+
minus 21198762
1199032+
+ 1198762) +
3
2(119903+
1198762
1199032+
+ 1198762)
minus (1199032
+
minus 21198762
1199032+
+ 1198762)119872120596
= 119864119862
+ (1199032
+
minus 21198762
1199032+
+ 1198762)119872120596
(33)
where the second term is the correction term The modifica-tion in extensive part of the energy can be obtained by addingthe term 2119864 in the above expression as
119864119864
=119903+
8(1199032
+
minus 21198762
1199032+
+ 1198762) minus
3
4(119903+
1198762
1199032+
+ 1198762) minus (
1199032
+
minus 21198762
1199032+
+ 1198762)119872120596
= 119864119864
minus (1199032
+
minus 21198762
1199032+
+ 1198762)119872120596
(34)
The extensive energy (16) can also be modified in the contextof KKW analysis as follows
119864119864
=119886
41205871198771198782
bh =120587119886
41198771199034
out =120587119886
41198771199034
+
(1 minus120596
119872) (35)
Inserting (18) (33) and (34) in modified CV formula that is
119878CFT =2120587119877
radic119886119887
radic119864119862
(2119864 minus 119864119862
) (36)
it follows that
119878CFT = 119878(((8119872119903+ (1199032
+
+ 1198762
) minus 1199032
+
(1199032
+
minus 21198762
)
+64
119903+
(1199032
+
+ 1198762
)1198722
120596 + 8119903+
(1199032
+
minus 21198762
)119872120596
minus8 (1199032
+
minus 21198762
)119872120596) + 121198762
1199032
+
+ 361198764
1199032
+
)
times [
[
(1199032
+
minus 21198762
)minus2
(4119872 (1199032+
+ 1198762) minus 120572) 120572
]
]
)
12
(37)
where 120572 = (2120587119903+
(1199032
+
minus 21198762
) + 31198762
119903+
) This gives the correc-tions in CV formula due to the effects of self-gravitation
The total energy is the combination of extensive andCasimir energies so 119864 gt 119864
119862
which implies that the term2119864 minus 119864
119862
is positive As the correction term is positive so themodified term 119864
119862
(2119864 minus 119864119862
) is greater in magnitude than theoriginal one The self-gravitational correction of CV formulafor Bardeen BH remains positive as the term inside squareroot is real and positive The above term can be expanded upto the first-order in 120596 as follows
119864119862
(2119864 minus 119864119862
) = 119864119862
(2119864 minus 119864119862
)
minus (119864119862
minus (2119864 minus 119864119862
)) (1199032
+
minus 21198762
1199032+
+ 1198762)119872120596
(38)
which relates the modified self-gravitational energy with theoriginal energy It is seen that the self-gravitational modifiedtemperature and entropy derived in the context of KKWanalysis of Bardeen BH are different from the Bekenstein-Hawking temperature and entropy respectively The BHentropy described by CV formula violates the holographicbound as 119878CFT gt 119878 gt 119878bh [39 40]
Advances in High Energy Physics 5
32 Self-Gravitational Corrections for Ayon Beato-Garcıa-Bronnikov Black Hole Now we consider ABGB BH tocalculate the self-gravitational corrections in the CV formulausing KKW analysis The corresponding self-gravitationalcorrections in 119879bh119878bh are
119879bh119878bh = 119879119867119878 minus120596
2119872(119903+
+ 119876Φ+
) (39)
Equation (29) gives the modified Casimir energy as
119864119862
=3
2119872 +
5
4(1198762
119903+
sec ℎ2 ( 1198762
2119898119903+
)) +2
1199032+
(119903+
+ 119876Φ+
)119872120596
= 119864119862
+2
1199032+
(119903+
+ 119876Φ+
)119872120596
(40)
where the second term is the correction term Additionallythe modified extensive energy can be written as
2119864 minus 119864119862
= 2119864 minus 119864119862
minus2
1199032+
(119903+
+ 119876Φ+
)119872120596
119864119864
= 119864119864
minus1
1199032+
(119903+
+ 119876Φ+
)119872120596
(41)
Inserting (27) (40) and (41) in modified CV formula weobtain the self-gravitational corrected CV formula for ABGBBH as follows
119878CFT
= 119878[
[
((3
8119872119903+
minus4
1199032+
(119903+
+ 119876Φ+
)1198722
120596
minus25
16
1198764
1199032+
sec ℎ4 ( 1198762
2119898119903+
))
times ((1
8119903+
minus5
8
1198764
1199032+
sec ℎ4 ( 1198762
2119898119903+
))
times (3
2119872 +
5
4
1198764
1199032+
sec ℎ4 ( 1198762
2119898119903+
)))
minus1
)
12
]
]
(42)
As the correction term (21199032
+
)(119903+
+119876Φ+
) gt 0 this implies that119864119862
(2119864 minus 119864119862
) gt 119864119862
(2119864 minus 119864119862
) The corresponding first-orderexpansion of the correction term is
119864119862
(2119864 minus 119864119862
) = 119864119862
(2119864 minus 119864119862
)
+2
1199032+
(119903+
+ 119876Φ+
) (2119864 minus 119864119862
)119872120596
(43)
This shows that the ABGB BH entropy described by CVformula also violates the holographic bound
4 Concluding Remarks
The description of BH entropy by CV formula and itsquantum corrections have attained much attention Therehave been a number of papers to find the semiclassicalcorrections in this formula for different BHs using varioustechniques such as self-gravitational effects [41] generalizeduncertainty principle [42] space noncommutativity [43] andquantum corrections [44] The self-gravitational correctionallows one to study across horizon physics and provides thesolution of information loss paradox
This paper is devoted to study this analysis forthe Bardeen and ABGB BHs which correspond to theSchwarzschild BH for zero charges Firstly we have evaluatedthe thermodynamical quantities such as electric potentialHawking temperature and Bekenstein-Hawking entropy ofboth of the BHs It is found that there exists unique eventhorizon whose entropy can be expressed in the form of CVformula indicating that this formula holds on both eventhorizons
Secondly we have used the self-gravitational effect toevaluate the temperature and entropy of these BHs inthe context of KKW analysis The corrections in the CVformula due to self-gravitational effects are evaluated bytaking dynamical background of the BH The correctionsare restricted up to the linear order in 120596 for the zerothorder the modified temperature corresponds to the Hawkingtemperature We have also found the corrections in all thequantities which can be modified under self-gravitationaleffect that is pure extensive energy and the Casimir energyThe electric potential charge spherical radius and totalenergy remain invariant under this effect It turns out that theself-gravitational correction term in modified CV formula ispositive for both BHs as the modified term is greater than theoriginal It is interesting tomention here that the positive self-gravitational corrections for regular BHs do not satisfy theinequality 119878CFT gt 119878 gt 119878bh which proves that the holographicbound is not universal
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the Higher EducationCommission Islamabad Pakistan for its financial supportthrough the Indigenous PhD Fellowship for 5K ScholarsPhase-II Batch-I
References
[1] J Michell ldquoOn the means of discovering the distance magni-tude ampc of the fixed stars in consequence of the diminution ofthe velocity of their light in case such a diminution should befound to take place in any of them and such other data shouldbe procured from observations as would be farther necessary
6 Advances in High Energy Physics
for that purposerdquoPhilosophical Transactions of the Royal Societyvol 74 pp 35ndash57 1784
[2] J AWheeler ldquoOur universe the known and the unknownrdquoTheAmerican Scientist vol 56 p 1 1968
[3] S W Hawking ldquoBlack hole explosionsrdquo Nature vol 248 no5443 pp 30ndash31 1974
[4] J B Hartle and S W Hawking ldquoPath-integral derivation ofblack-hole radiancerdquo Physical Review D vol 13 no 8 pp 2188ndash2203 1976
[5] Q Q Jiang and X Cai ldquoRemarks on self-interaction correctionto black hole radiationrdquo Journal of High Energy Physics vol2003 no 11 article 110 2009
[6] P Kraus and F Wilczek ldquoSelf-interaction correction to blackhole radiancerdquo Nuclear Physics B vol 433 no 2 pp 403ndash4201995
[7] P Kraus and F Wilczek ldquoEffect of self-interaction on chargedblack hole radiancerdquo Nuclear Physics B vol 437 no 1 pp 231ndash242 1995
[8] P Painleve ldquoLamecanique classique et la theorie de la relativiterdquoComptes Rendus de lrsquoAcademie des Sciences vol 173 pp 677ndash680 1921
[9] E C Vagenas ldquoGeneralization of the KKW analysis for blackhole radiationrdquo Physics Letters B vol 559 no 1-2 pp 65ndash732003
[10] M K Parikh and F Wilczek ldquoHawking radiation as tunnelingrdquoPhysical Review Letters vol 85 no 24 pp 5042ndash5045 2000
[11] S Hemming and E Keski-Vakkuri ldquoHawking radiation fromAdS black holesrdquo Physical Review D vol 64 no 4 Article ID044006 8 pages 2001
[12] Y Kwon ldquoHawking radiation in AdS2
black holerdquo Il NuovoCimento della Societa Italiana di Fisica B Serie 12 vol 115 no 4pp 469ndash471 2000
[13] E C Vagenas ldquoAre extremal 2D black holes really frozenrdquoPhysics Letters B vol 503 no 3-4 pp 399ndash403 2001
[14] E C Vagenas ldquoTwo-dimensional dilatonic black holes andHawking radiationrdquo Modern Physics Letters A Particles andFields Gravitation Cosmology Nuclear Physics vol 17 no 10pp 609ndash618 2002
[15] E C Vagenas ldquoSemiclassical corrections to the Bekenstein-Hawking entropy of the BTZ black hole via self-gravitationrdquoPhysics Letters B vol 533 no 3-4 pp 302ndash306 2002
[16] A J M Medved ldquoRadiation via tunnelling in the charged BTZblack holerdquo Classical and Quantum Gravity vol 19 no 3 pp589ndash598 2002
[17] A J M Medved ldquoRadiation via tunneling from a de Sittercosmological horizonrdquo Physical ReviewD vol 66 no 12 ArticleID 124009 7 pages 2002
[18] L Susskind ldquoTheworld as a hologramrdquo Journal ofMathematicalPhysics vol 36 no 11 pp 6377ndash6396 1995
[19] E Verlinde ldquoOn the holographic principle in a radiationdominateduniverserdquo httparxivorgabshepth0008140
[20] D Birmingham and S Mokhtari ldquoThe Cardy-Verlinde formulaand Taub-bolt-AdS spacetimesrdquo Physics Letters B vol 508 no3-4 pp 365ndash368 2001
[21] D Klemm A C Petkou and G Siopsis ldquoEntropy boundsmonotonicity properties and scaling in CFTsrdquo Nuclear PhysicsB vol 601 no 1-2 pp 380ndash394 2001
[22] B Wang E Abdalla and R-K Su ldquoRelating Friedmannequation to Cardy formula in universes with cosmologicalconstantrdquo Physics Letters B vol 503 no 3-4 pp 394ndash398 2001
[23] M R Setare ldquoThe Cardy-Verlinde formula and entropy oftopological Reissner-Nordstrom black holes in de Sitter spacesrdquoModern Physics Letters A Particles and Fields GravitationCosmology Nuclear Physics vol 17 no 32 pp 2089ndash2094 2002
[24] M R Setare andM B Altaie ldquoTheCardy-Verlinde formula andentropy of topological Kerr-Newman black holes in de Sitterspacesrdquo The European Physical Journal C Particles and Fieldsvol 30 no 2 pp 273ndash277 2003
[25] M R Setare and R Mansouri ldquoHolographic thermodynamicson the brane in topological Reissner-Nordstromde Sitter spacerdquoInternational Journal of Modern Physics A vol 18 no 24 pp4443ndash4450 2003
[26] C O Lee ldquoCardy-Verlinde formula in Taub-NUTbolt-(A)dSspacerdquo Physics Letters B vol 670 no 2 pp 146ndash149 2008
[27] A J M Medved ldquoQuantum-corrected Cardy entropy forgeneric (1 + 1)-dimensional gravityrdquo Classical and QuantumGravity vol 19 no 9 pp 2503ndash2513 2002
[28] SMukherji and S S Pal ldquoLogarithmic corrections to black holeentropy and AdSCFT correspondencerdquo Journal of High EnergyPhysics vol 2002 no 5 article 026 2002
[29] J E Lidsey S Nojiri S D Odintsov and S OgushildquoThe AdSCFT correspondence and logarithmic correctionsto braneworld cosmology and the Cardy-Verlinde formulardquoPhysics Letters B vol 544 no 3-4 pp 337ndash345 2002
[30] S Carlip ldquoLogarithmic corrections to black hole entropy fromthe Cardy formulardquo Classical and Quantum Gravity vol 17 no20 pp 4175ndash4186 2000
[31] M R Setare and E C Vagenas ldquoCardy-Verlinde formula andAchucarro-Ortiz black holerdquo Physical Review D vol 68 no 6Article ID 064014 5 pages 2003
[32] MR Setare andECVagenas ldquoSelf-gravitational corrections tothe Cardy-Verlinde formula of the Achucarro-Ortiz black holerdquoPhysics Letters B vol 584 no 1-2 pp 127ndash132 2004
[33] M R Setare and M Jamil ldquoThe Cardy-Verlinde formula andentropy of the charged rotating BTZ black holerdquo Physics LettersB vol 681 no 5 pp 469ndash471 2009
[34] F Darabi M Jamil and M R Setare ldquoSelf-gravitationalcorrections to theCardy-Verlinde formula of chargedBTZblackholerdquoModern Physics Letters A Particles and Fields GravitationCosmology Nuclear Physics vol 26 no 14 pp 1047ndash1057 2011
[35] G Abbas ldquoCardy-Verlinde formula of noncommutativeSchwarzschild black holerdquo Advances in High Energy Physicsvol 2014 Article ID 306256 4 pages 2014
[36] J Bardeen ldquoNon-singular general relativistic gravitational col-lapserdquo in Proceedings of the 5th International Conference onGravitation and the Theory of Relativity Tbilisi Unversty PressTbilisi Georgia September 1968
[37] E Ayon-Beato and A Garcıa ldquoNew regular black hole solutionfrom nonlinear electrodynamicsrdquo Physics Letters B vol 464 no1-2 pp 25ndash29 1999
[38] K A Bronnikov ldquoComment on lsquoregular black hole in gen-eral relativity coupled to nonlinear electrodynamicsrsquordquo PhysicalReview Letters vol 85 p 4641 2000
[39] R Bousso ldquoA covariant entropy conjecturerdquo Journal of HighEnergy Physics vol 1999 no 7 article 004 1999
[40] G rsquot Hooft ldquoDimensional reduction in quantumgravityrdquohttparxivorgabsgr-qc9310026
[41] M R Setare and E C Vagenas ldquoSelf-gravitational correctionsto the Cardy-Verlinde formula and the FRW brane cosmologyin SdS
5
bulkrdquo International Journal of Modern Physics A vol 20no 30 pp 7219ndash7232 2005
Advances in High Energy Physics 7
[42] M R Setare ldquoThe generalized uncertainty principle and cor-rections to the Cardy-Verlinde formula in SAdS
5
black holesrdquoInternational Journal of Modern Physics A vol 21 no 6 p 13252006
[43] M R Setare ldquoSpace noncommutativity corrections to theCardy-Verlinde formulardquo International Journal of ModernPhysics A vol 21 no 13-14 p 3007 2006
[44] M Sharif and W Javed ldquoQuantum corrections for ABGB blackholerdquoAstrophysics and Space Science vol 337 no 1 pp 335ndash3412012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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FluidsJournal of
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Advances in Condensed Matter Physics
OpticsInternational Journal of
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Superconductivity
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Statistical MechanicsInternational Journal of
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Physics Research International
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Soft MatterJournal of
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Journal of
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ThermodynamicsJournal of
![Page 3: Research Article Cardy-Verlinde Formula and Its Self ...downloads.hindawi.com/journals/ahep/2014/926589.pdf · e generalized Cardy-Verlinde formula is de ned as [ ] CFT = 2 % & '*](https://reader035.vdocuments.net/reader035/viewer/2022071219/6056d064e08cc116434a5561/html5/thumbnails/3.jpg)
Advances in High Energy Physics 3
which provides 119899-dimensional Casimir energy in the follow-ing form
119864119862
= 119899 (119864 + 119875119881 minus 119879119878 minus Φ+
119876) (11)where pressure is defined as119875 = 119864119899119881The product of energyand radius (ER) is independent of volume due to conformalinvariance which is satisfied by both 119864
119864
and 119864119862
Hence bothenergies can be expressed in terms of 119877 and 119878 in arbitrarydimensions as follows [19]
119864119862
=119887
21205871198771198781minus1119899
119864119864
=119886
41205871198771198781+1119899
(12)
Taking 119899 = 1 and using the above expressions of 119875 119879119867
119878 Φ+
in (11) the Casimir energy takes the form
119864119862
= 2119864 minus119903+
4(1199032
+
minus 21198762
1199032+
+ 1198762) + (
31198762
119903+
1199032+
+ 1198762) (13)
Using mass energy relation (119864 = 119872) this reduces to
119864119862
= 2119872 minus119903+
4(1199032
+
minus 21198762
1199032+
+ 1198762) + (
31198762
119903+
1199032+
+ 1198762) (14)
The extensive energy is found through (9) and (13) as
2119864119864
= 2119864 minus 119864119862
= minus119903+
4(1199032
+
minus 21198762
1199032+
+ 1198762) + (
31198762
119903+
1199032+
+ 1198762) (15)
Using (7) the generalized formulas of Casimir and extensiveenergy in one dimension can be written as
119864119862
=119887
2120587119877 119864
119864
=120587119886
41198771199034
+
(16)
Comparing the Casimir as well as extensive energy given in(14)ndash(16) we obtain two different spherical radii as
119877 = 1205871198861199034
+
[
(1199032
+
+ 1198762
)
2120587119903+
(1199032+
minus 21198762) minus 31198762119903+
]
119877 =119887
120587
times [
(1199032
+
+ 1198762
)
4119872 (1199032+
+ 1198762) minus 2119903+
(1199032+
minus 21198762) minus 31198762119903+
]
(17)
Multiplication of the above two equations provides119877
= (119886119887119903+
(1199032
+
+ 1198762
)2
times ((2120587119903+
(1199032
+
minus 21198762
) minus 31198762
119903+
)
times (4119872(1199032
+
+ 1198762
) minus 2119903+
(1199032
+
minus 21198762
) minus 31198762
119903+
))minus1
)
12
(18)Substituting (14) (15) and (18) in CV formula the final resultis
119878CFT = 1205871199032
+
= 119878 (19)This shows that the entropy of Bardeen BH can be expressedin terms of CV formula in the context of CFT
22 Entropy of Ayon Beato-Garcıa-Bronnikov Black HoleAyon Beato and Garcıa [37] and Bronnikov [38] proposed anonsingular BH by solving the system of equations coupledwith nonlinear electrodynamics and gravity For this BH themass function is given as
119872(119903) = 119898[1 minus tanh( 1198762
2119898119903)] (20)
which also reduces to the Schwarzschild BH for 119876 = 0 Theevent horizon has the following form
119903+
= 2119872(119903+
) = 2119898[1 minus tanh( 1198762
2119898119903+
)] (21)
The corresponding electric potential Hawking temperatureand entropy turn out to be
Φ+
=120597119872
120597119876
10038161003816100381610038161003816100381610038161003816119903=119903
+
= minus119876
119903+
sec ℎ2 ( 1198762
2119898119903+
)
119879119867
=1
4120587
119889119860
119889119903
10038161003816100381610038161003816100381610038161003816119903=119903+
=1
41205871199032+
(119903+
+ 119876Φ+
)
119878 =A
4= 1205871199032
+
= 120587(2119898[1 minus tanh( 1198762
2119898119903+
)])
2
(22)
Using (11) the Casimir energy can be written as
119864119862
= 2119864 minus1
4119903+
+5
4(1198762
119903+
sec ℎ2 ( 1198762
2119898119903+
)) (23)
For 119864 = 119872 this reduces to
119864119862
=3
2119872 +
5
4(1198762
119903+
sec ℎ2 ( 1198762
2119898119903+
)) (24)
The extensive energy is given as
2119864119864
= 2119864 minus 119864119862
=1
4119903+
minus5
4(1198762
119903+
sec ℎ2 ( 1198762
2119898119903+
)) (25)
Comparison of (24) and (25) with (16) leads to
119877 =120587119886
41199034
+
[1
(18) 119903+
minus (58) ((1198762119903+
) sec ℎ2 (11987622119898119903+
))]
119877 =119887
2120587[
1
(32)119872 + (54) ((1198762119903+
) sec ℎ2 (11987622119898119903+
))]
(26)
yielding the unique radius
119877 =radic119886119887
2radic21199032
+
times [1 times ((3
2119872 +
5
4(1198762
119903+
sec ℎ2 ( 1198762
2119898119903+
)))
times(1
8119903+
minus5
8(1198762
119903+
sec ℎ2 ( 1198762
2119898119903+
))))
minus12
]
(27)
4 Advances in High Energy Physics
Inserting (24)ndash(27) in (8) we find that the entropy of CFT isequivalent to the entropy of ABGB BH as
119878CFT = 1205871199032
+
= 119878 (28)
3 Self-Gravitational Corrections tothe Cardy-Verlinde Formula
In this section we evaluate the self-gravitational correctionsin the CV formula for both BHs using KKW analysis Thisleads to modifying all the quantities in the above sectionunder self-gravitational effects
31 Self-Gravitational Corrections for Bardeen Black HoleHere we evaluate the self-gravitational corrections in CVformula for Bardeen BH The modified Casimir energy (119864
119862
)
is defined as
119864119862
= 2119864 minus 119879bh119878bh minus 119876bhΦbh (29)
where 119879bh and 119878bh stand for modified temperature andentropy respectively The third term of (11) is modified to119879bh119878bh whereas the total energy charge radius of the sphereand electric potential of BH remain invariant under thiseffect In order to evaluate the corrections in BH temperatureand entropy we use the KKW analysis In this analysis thetotalmass is kept fixed while BHmass is assumed to fluctuatebecause a shell of energy 120596 radiates massless particles andhence BH mass reduces to 119872 minus 120596 This shell energy doesnot contain charge due to which electric potential remainsthe same that is 119876bhΦbh = 119876Φ+ Applying this analysis weobtain a relationship between BH entropy and Bekenstein-Hawking entropy as
119878bh = 119878 minus 41205871198722
[1 minus (1 minus120596
119872)
2
] (30)
The corresponding self-gravitational corrected temperature isgiven by
119879bh =120596
41205871198722[1 minus (1 minus
120596
119872)
2
]
minus1
(31)
which reduces to the Hawking temperature by consideringcorrections up to first-order in 120596
Thus the self-gravitational corrections in 119879bh119878bh take theform
119879bh119878bh = 119879119867119878 minus2
119903+
(1199032
+
minus 21198762
1199032+
+ 1198762)119872120596 (32)
Substituting this value in (29) the modified Casimir energyturns out to be
119864119862
= 2119872 minus119903+
4(1199032
+
minus 21198762
1199032+
+ 1198762) +
3
2(119903+
1198762
1199032+
+ 1198762)
minus (1199032
+
minus 21198762
1199032+
+ 1198762)119872120596
= 119864119862
+ (1199032
+
minus 21198762
1199032+
+ 1198762)119872120596
(33)
where the second term is the correction term The modifica-tion in extensive part of the energy can be obtained by addingthe term 2119864 in the above expression as
119864119864
=119903+
8(1199032
+
minus 21198762
1199032+
+ 1198762) minus
3
4(119903+
1198762
1199032+
+ 1198762) minus (
1199032
+
minus 21198762
1199032+
+ 1198762)119872120596
= 119864119864
minus (1199032
+
minus 21198762
1199032+
+ 1198762)119872120596
(34)
The extensive energy (16) can also be modified in the contextof KKW analysis as follows
119864119864
=119886
41205871198771198782
bh =120587119886
41198771199034
out =120587119886
41198771199034
+
(1 minus120596
119872) (35)
Inserting (18) (33) and (34) in modified CV formula that is
119878CFT =2120587119877
radic119886119887
radic119864119862
(2119864 minus 119864119862
) (36)
it follows that
119878CFT = 119878(((8119872119903+ (1199032
+
+ 1198762
) minus 1199032
+
(1199032
+
minus 21198762
)
+64
119903+
(1199032
+
+ 1198762
)1198722
120596 + 8119903+
(1199032
+
minus 21198762
)119872120596
minus8 (1199032
+
minus 21198762
)119872120596) + 121198762
1199032
+
+ 361198764
1199032
+
)
times [
[
(1199032
+
minus 21198762
)minus2
(4119872 (1199032+
+ 1198762) minus 120572) 120572
]
]
)
12
(37)
where 120572 = (2120587119903+
(1199032
+
minus 21198762
) + 31198762
119903+
) This gives the correc-tions in CV formula due to the effects of self-gravitation
The total energy is the combination of extensive andCasimir energies so 119864 gt 119864
119862
which implies that the term2119864 minus 119864
119862
is positive As the correction term is positive so themodified term 119864
119862
(2119864 minus 119864119862
) is greater in magnitude than theoriginal one The self-gravitational correction of CV formulafor Bardeen BH remains positive as the term inside squareroot is real and positive The above term can be expanded upto the first-order in 120596 as follows
119864119862
(2119864 minus 119864119862
) = 119864119862
(2119864 minus 119864119862
)
minus (119864119862
minus (2119864 minus 119864119862
)) (1199032
+
minus 21198762
1199032+
+ 1198762)119872120596
(38)
which relates the modified self-gravitational energy with theoriginal energy It is seen that the self-gravitational modifiedtemperature and entropy derived in the context of KKWanalysis of Bardeen BH are different from the Bekenstein-Hawking temperature and entropy respectively The BHentropy described by CV formula violates the holographicbound as 119878CFT gt 119878 gt 119878bh [39 40]
Advances in High Energy Physics 5
32 Self-Gravitational Corrections for Ayon Beato-Garcıa-Bronnikov Black Hole Now we consider ABGB BH tocalculate the self-gravitational corrections in the CV formulausing KKW analysis The corresponding self-gravitationalcorrections in 119879bh119878bh are
119879bh119878bh = 119879119867119878 minus120596
2119872(119903+
+ 119876Φ+
) (39)
Equation (29) gives the modified Casimir energy as
119864119862
=3
2119872 +
5
4(1198762
119903+
sec ℎ2 ( 1198762
2119898119903+
)) +2
1199032+
(119903+
+ 119876Φ+
)119872120596
= 119864119862
+2
1199032+
(119903+
+ 119876Φ+
)119872120596
(40)
where the second term is the correction term Additionallythe modified extensive energy can be written as
2119864 minus 119864119862
= 2119864 minus 119864119862
minus2
1199032+
(119903+
+ 119876Φ+
)119872120596
119864119864
= 119864119864
minus1
1199032+
(119903+
+ 119876Φ+
)119872120596
(41)
Inserting (27) (40) and (41) in modified CV formula weobtain the self-gravitational corrected CV formula for ABGBBH as follows
119878CFT
= 119878[
[
((3
8119872119903+
minus4
1199032+
(119903+
+ 119876Φ+
)1198722
120596
minus25
16
1198764
1199032+
sec ℎ4 ( 1198762
2119898119903+
))
times ((1
8119903+
minus5
8
1198764
1199032+
sec ℎ4 ( 1198762
2119898119903+
))
times (3
2119872 +
5
4
1198764
1199032+
sec ℎ4 ( 1198762
2119898119903+
)))
minus1
)
12
]
]
(42)
As the correction term (21199032
+
)(119903+
+119876Φ+
) gt 0 this implies that119864119862
(2119864 minus 119864119862
) gt 119864119862
(2119864 minus 119864119862
) The corresponding first-orderexpansion of the correction term is
119864119862
(2119864 minus 119864119862
) = 119864119862
(2119864 minus 119864119862
)
+2
1199032+
(119903+
+ 119876Φ+
) (2119864 minus 119864119862
)119872120596
(43)
This shows that the ABGB BH entropy described by CVformula also violates the holographic bound
4 Concluding Remarks
The description of BH entropy by CV formula and itsquantum corrections have attained much attention Therehave been a number of papers to find the semiclassicalcorrections in this formula for different BHs using varioustechniques such as self-gravitational effects [41] generalizeduncertainty principle [42] space noncommutativity [43] andquantum corrections [44] The self-gravitational correctionallows one to study across horizon physics and provides thesolution of information loss paradox
This paper is devoted to study this analysis forthe Bardeen and ABGB BHs which correspond to theSchwarzschild BH for zero charges Firstly we have evaluatedthe thermodynamical quantities such as electric potentialHawking temperature and Bekenstein-Hawking entropy ofboth of the BHs It is found that there exists unique eventhorizon whose entropy can be expressed in the form of CVformula indicating that this formula holds on both eventhorizons
Secondly we have used the self-gravitational effect toevaluate the temperature and entropy of these BHs inthe context of KKW analysis The corrections in the CVformula due to self-gravitational effects are evaluated bytaking dynamical background of the BH The correctionsare restricted up to the linear order in 120596 for the zerothorder the modified temperature corresponds to the Hawkingtemperature We have also found the corrections in all thequantities which can be modified under self-gravitationaleffect that is pure extensive energy and the Casimir energyThe electric potential charge spherical radius and totalenergy remain invariant under this effect It turns out that theself-gravitational correction term in modified CV formula ispositive for both BHs as the modified term is greater than theoriginal It is interesting tomention here that the positive self-gravitational corrections for regular BHs do not satisfy theinequality 119878CFT gt 119878 gt 119878bh which proves that the holographicbound is not universal
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the Higher EducationCommission Islamabad Pakistan for its financial supportthrough the Indigenous PhD Fellowship for 5K ScholarsPhase-II Batch-I
References
[1] J Michell ldquoOn the means of discovering the distance magni-tude ampc of the fixed stars in consequence of the diminution ofthe velocity of their light in case such a diminution should befound to take place in any of them and such other data shouldbe procured from observations as would be farther necessary
6 Advances in High Energy Physics
for that purposerdquoPhilosophical Transactions of the Royal Societyvol 74 pp 35ndash57 1784
[2] J AWheeler ldquoOur universe the known and the unknownrdquoTheAmerican Scientist vol 56 p 1 1968
[3] S W Hawking ldquoBlack hole explosionsrdquo Nature vol 248 no5443 pp 30ndash31 1974
[4] J B Hartle and S W Hawking ldquoPath-integral derivation ofblack-hole radiancerdquo Physical Review D vol 13 no 8 pp 2188ndash2203 1976
[5] Q Q Jiang and X Cai ldquoRemarks on self-interaction correctionto black hole radiationrdquo Journal of High Energy Physics vol2003 no 11 article 110 2009
[6] P Kraus and F Wilczek ldquoSelf-interaction correction to blackhole radiancerdquo Nuclear Physics B vol 433 no 2 pp 403ndash4201995
[7] P Kraus and F Wilczek ldquoEffect of self-interaction on chargedblack hole radiancerdquo Nuclear Physics B vol 437 no 1 pp 231ndash242 1995
[8] P Painleve ldquoLamecanique classique et la theorie de la relativiterdquoComptes Rendus de lrsquoAcademie des Sciences vol 173 pp 677ndash680 1921
[9] E C Vagenas ldquoGeneralization of the KKW analysis for blackhole radiationrdquo Physics Letters B vol 559 no 1-2 pp 65ndash732003
[10] M K Parikh and F Wilczek ldquoHawking radiation as tunnelingrdquoPhysical Review Letters vol 85 no 24 pp 5042ndash5045 2000
[11] S Hemming and E Keski-Vakkuri ldquoHawking radiation fromAdS black holesrdquo Physical Review D vol 64 no 4 Article ID044006 8 pages 2001
[12] Y Kwon ldquoHawking radiation in AdS2
black holerdquo Il NuovoCimento della Societa Italiana di Fisica B Serie 12 vol 115 no 4pp 469ndash471 2000
[13] E C Vagenas ldquoAre extremal 2D black holes really frozenrdquoPhysics Letters B vol 503 no 3-4 pp 399ndash403 2001
[14] E C Vagenas ldquoTwo-dimensional dilatonic black holes andHawking radiationrdquo Modern Physics Letters A Particles andFields Gravitation Cosmology Nuclear Physics vol 17 no 10pp 609ndash618 2002
[15] E C Vagenas ldquoSemiclassical corrections to the Bekenstein-Hawking entropy of the BTZ black hole via self-gravitationrdquoPhysics Letters B vol 533 no 3-4 pp 302ndash306 2002
[16] A J M Medved ldquoRadiation via tunnelling in the charged BTZblack holerdquo Classical and Quantum Gravity vol 19 no 3 pp589ndash598 2002
[17] A J M Medved ldquoRadiation via tunneling from a de Sittercosmological horizonrdquo Physical ReviewD vol 66 no 12 ArticleID 124009 7 pages 2002
[18] L Susskind ldquoTheworld as a hologramrdquo Journal ofMathematicalPhysics vol 36 no 11 pp 6377ndash6396 1995
[19] E Verlinde ldquoOn the holographic principle in a radiationdominateduniverserdquo httparxivorgabshepth0008140
[20] D Birmingham and S Mokhtari ldquoThe Cardy-Verlinde formulaand Taub-bolt-AdS spacetimesrdquo Physics Letters B vol 508 no3-4 pp 365ndash368 2001
[21] D Klemm A C Petkou and G Siopsis ldquoEntropy boundsmonotonicity properties and scaling in CFTsrdquo Nuclear PhysicsB vol 601 no 1-2 pp 380ndash394 2001
[22] B Wang E Abdalla and R-K Su ldquoRelating Friedmannequation to Cardy formula in universes with cosmologicalconstantrdquo Physics Letters B vol 503 no 3-4 pp 394ndash398 2001
[23] M R Setare ldquoThe Cardy-Verlinde formula and entropy oftopological Reissner-Nordstrom black holes in de Sitter spacesrdquoModern Physics Letters A Particles and Fields GravitationCosmology Nuclear Physics vol 17 no 32 pp 2089ndash2094 2002
[24] M R Setare andM B Altaie ldquoTheCardy-Verlinde formula andentropy of topological Kerr-Newman black holes in de Sitterspacesrdquo The European Physical Journal C Particles and Fieldsvol 30 no 2 pp 273ndash277 2003
[25] M R Setare and R Mansouri ldquoHolographic thermodynamicson the brane in topological Reissner-Nordstromde Sitter spacerdquoInternational Journal of Modern Physics A vol 18 no 24 pp4443ndash4450 2003
[26] C O Lee ldquoCardy-Verlinde formula in Taub-NUTbolt-(A)dSspacerdquo Physics Letters B vol 670 no 2 pp 146ndash149 2008
[27] A J M Medved ldquoQuantum-corrected Cardy entropy forgeneric (1 + 1)-dimensional gravityrdquo Classical and QuantumGravity vol 19 no 9 pp 2503ndash2513 2002
[28] SMukherji and S S Pal ldquoLogarithmic corrections to black holeentropy and AdSCFT correspondencerdquo Journal of High EnergyPhysics vol 2002 no 5 article 026 2002
[29] J E Lidsey S Nojiri S D Odintsov and S OgushildquoThe AdSCFT correspondence and logarithmic correctionsto braneworld cosmology and the Cardy-Verlinde formulardquoPhysics Letters B vol 544 no 3-4 pp 337ndash345 2002
[30] S Carlip ldquoLogarithmic corrections to black hole entropy fromthe Cardy formulardquo Classical and Quantum Gravity vol 17 no20 pp 4175ndash4186 2000
[31] M R Setare and E C Vagenas ldquoCardy-Verlinde formula andAchucarro-Ortiz black holerdquo Physical Review D vol 68 no 6Article ID 064014 5 pages 2003
[32] MR Setare andECVagenas ldquoSelf-gravitational corrections tothe Cardy-Verlinde formula of the Achucarro-Ortiz black holerdquoPhysics Letters B vol 584 no 1-2 pp 127ndash132 2004
[33] M R Setare and M Jamil ldquoThe Cardy-Verlinde formula andentropy of the charged rotating BTZ black holerdquo Physics LettersB vol 681 no 5 pp 469ndash471 2009
[34] F Darabi M Jamil and M R Setare ldquoSelf-gravitationalcorrections to theCardy-Verlinde formula of chargedBTZblackholerdquoModern Physics Letters A Particles and Fields GravitationCosmology Nuclear Physics vol 26 no 14 pp 1047ndash1057 2011
[35] G Abbas ldquoCardy-Verlinde formula of noncommutativeSchwarzschild black holerdquo Advances in High Energy Physicsvol 2014 Article ID 306256 4 pages 2014
[36] J Bardeen ldquoNon-singular general relativistic gravitational col-lapserdquo in Proceedings of the 5th International Conference onGravitation and the Theory of Relativity Tbilisi Unversty PressTbilisi Georgia September 1968
[37] E Ayon-Beato and A Garcıa ldquoNew regular black hole solutionfrom nonlinear electrodynamicsrdquo Physics Letters B vol 464 no1-2 pp 25ndash29 1999
[38] K A Bronnikov ldquoComment on lsquoregular black hole in gen-eral relativity coupled to nonlinear electrodynamicsrsquordquo PhysicalReview Letters vol 85 p 4641 2000
[39] R Bousso ldquoA covariant entropy conjecturerdquo Journal of HighEnergy Physics vol 1999 no 7 article 004 1999
[40] G rsquot Hooft ldquoDimensional reduction in quantumgravityrdquohttparxivorgabsgr-qc9310026
[41] M R Setare and E C Vagenas ldquoSelf-gravitational correctionsto the Cardy-Verlinde formula and the FRW brane cosmologyin SdS
5
bulkrdquo International Journal of Modern Physics A vol 20no 30 pp 7219ndash7232 2005
Advances in High Energy Physics 7
[42] M R Setare ldquoThe generalized uncertainty principle and cor-rections to the Cardy-Verlinde formula in SAdS
5
black holesrdquoInternational Journal of Modern Physics A vol 21 no 6 p 13252006
[43] M R Setare ldquoSpace noncommutativity corrections to theCardy-Verlinde formulardquo International Journal of ModernPhysics A vol 21 no 13-14 p 3007 2006
[44] M Sharif and W Javed ldquoQuantum corrections for ABGB blackholerdquoAstrophysics and Space Science vol 337 no 1 pp 335ndash3412012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
![Page 4: Research Article Cardy-Verlinde Formula and Its Self ...downloads.hindawi.com/journals/ahep/2014/926589.pdf · e generalized Cardy-Verlinde formula is de ned as [ ] CFT = 2 % & '*](https://reader035.vdocuments.net/reader035/viewer/2022071219/6056d064e08cc116434a5561/html5/thumbnails/4.jpg)
4 Advances in High Energy Physics
Inserting (24)ndash(27) in (8) we find that the entropy of CFT isequivalent to the entropy of ABGB BH as
119878CFT = 1205871199032
+
= 119878 (28)
3 Self-Gravitational Corrections tothe Cardy-Verlinde Formula
In this section we evaluate the self-gravitational correctionsin the CV formula for both BHs using KKW analysis Thisleads to modifying all the quantities in the above sectionunder self-gravitational effects
31 Self-Gravitational Corrections for Bardeen Black HoleHere we evaluate the self-gravitational corrections in CVformula for Bardeen BH The modified Casimir energy (119864
119862
)
is defined as
119864119862
= 2119864 minus 119879bh119878bh minus 119876bhΦbh (29)
where 119879bh and 119878bh stand for modified temperature andentropy respectively The third term of (11) is modified to119879bh119878bh whereas the total energy charge radius of the sphereand electric potential of BH remain invariant under thiseffect In order to evaluate the corrections in BH temperatureand entropy we use the KKW analysis In this analysis thetotalmass is kept fixed while BHmass is assumed to fluctuatebecause a shell of energy 120596 radiates massless particles andhence BH mass reduces to 119872 minus 120596 This shell energy doesnot contain charge due to which electric potential remainsthe same that is 119876bhΦbh = 119876Φ+ Applying this analysis weobtain a relationship between BH entropy and Bekenstein-Hawking entropy as
119878bh = 119878 minus 41205871198722
[1 minus (1 minus120596
119872)
2
] (30)
The corresponding self-gravitational corrected temperature isgiven by
119879bh =120596
41205871198722[1 minus (1 minus
120596
119872)
2
]
minus1
(31)
which reduces to the Hawking temperature by consideringcorrections up to first-order in 120596
Thus the self-gravitational corrections in 119879bh119878bh take theform
119879bh119878bh = 119879119867119878 minus2
119903+
(1199032
+
minus 21198762
1199032+
+ 1198762)119872120596 (32)
Substituting this value in (29) the modified Casimir energyturns out to be
119864119862
= 2119872 minus119903+
4(1199032
+
minus 21198762
1199032+
+ 1198762) +
3
2(119903+
1198762
1199032+
+ 1198762)
minus (1199032
+
minus 21198762
1199032+
+ 1198762)119872120596
= 119864119862
+ (1199032
+
minus 21198762
1199032+
+ 1198762)119872120596
(33)
where the second term is the correction term The modifica-tion in extensive part of the energy can be obtained by addingthe term 2119864 in the above expression as
119864119864
=119903+
8(1199032
+
minus 21198762
1199032+
+ 1198762) minus
3
4(119903+
1198762
1199032+
+ 1198762) minus (
1199032
+
minus 21198762
1199032+
+ 1198762)119872120596
= 119864119864
minus (1199032
+
minus 21198762
1199032+
+ 1198762)119872120596
(34)
The extensive energy (16) can also be modified in the contextof KKW analysis as follows
119864119864
=119886
41205871198771198782
bh =120587119886
41198771199034
out =120587119886
41198771199034
+
(1 minus120596
119872) (35)
Inserting (18) (33) and (34) in modified CV formula that is
119878CFT =2120587119877
radic119886119887
radic119864119862
(2119864 minus 119864119862
) (36)
it follows that
119878CFT = 119878(((8119872119903+ (1199032
+
+ 1198762
) minus 1199032
+
(1199032
+
minus 21198762
)
+64
119903+
(1199032
+
+ 1198762
)1198722
120596 + 8119903+
(1199032
+
minus 21198762
)119872120596
minus8 (1199032
+
minus 21198762
)119872120596) + 121198762
1199032
+
+ 361198764
1199032
+
)
times [
[
(1199032
+
minus 21198762
)minus2
(4119872 (1199032+
+ 1198762) minus 120572) 120572
]
]
)
12
(37)
where 120572 = (2120587119903+
(1199032
+
minus 21198762
) + 31198762
119903+
) This gives the correc-tions in CV formula due to the effects of self-gravitation
The total energy is the combination of extensive andCasimir energies so 119864 gt 119864
119862
which implies that the term2119864 minus 119864
119862
is positive As the correction term is positive so themodified term 119864
119862
(2119864 minus 119864119862
) is greater in magnitude than theoriginal one The self-gravitational correction of CV formulafor Bardeen BH remains positive as the term inside squareroot is real and positive The above term can be expanded upto the first-order in 120596 as follows
119864119862
(2119864 minus 119864119862
) = 119864119862
(2119864 minus 119864119862
)
minus (119864119862
minus (2119864 minus 119864119862
)) (1199032
+
minus 21198762
1199032+
+ 1198762)119872120596
(38)
which relates the modified self-gravitational energy with theoriginal energy It is seen that the self-gravitational modifiedtemperature and entropy derived in the context of KKWanalysis of Bardeen BH are different from the Bekenstein-Hawking temperature and entropy respectively The BHentropy described by CV formula violates the holographicbound as 119878CFT gt 119878 gt 119878bh [39 40]
Advances in High Energy Physics 5
32 Self-Gravitational Corrections for Ayon Beato-Garcıa-Bronnikov Black Hole Now we consider ABGB BH tocalculate the self-gravitational corrections in the CV formulausing KKW analysis The corresponding self-gravitationalcorrections in 119879bh119878bh are
119879bh119878bh = 119879119867119878 minus120596
2119872(119903+
+ 119876Φ+
) (39)
Equation (29) gives the modified Casimir energy as
119864119862
=3
2119872 +
5
4(1198762
119903+
sec ℎ2 ( 1198762
2119898119903+
)) +2
1199032+
(119903+
+ 119876Φ+
)119872120596
= 119864119862
+2
1199032+
(119903+
+ 119876Φ+
)119872120596
(40)
where the second term is the correction term Additionallythe modified extensive energy can be written as
2119864 minus 119864119862
= 2119864 minus 119864119862
minus2
1199032+
(119903+
+ 119876Φ+
)119872120596
119864119864
= 119864119864
minus1
1199032+
(119903+
+ 119876Φ+
)119872120596
(41)
Inserting (27) (40) and (41) in modified CV formula weobtain the self-gravitational corrected CV formula for ABGBBH as follows
119878CFT
= 119878[
[
((3
8119872119903+
minus4
1199032+
(119903+
+ 119876Φ+
)1198722
120596
minus25
16
1198764
1199032+
sec ℎ4 ( 1198762
2119898119903+
))
times ((1
8119903+
minus5
8
1198764
1199032+
sec ℎ4 ( 1198762
2119898119903+
))
times (3
2119872 +
5
4
1198764
1199032+
sec ℎ4 ( 1198762
2119898119903+
)))
minus1
)
12
]
]
(42)
As the correction term (21199032
+
)(119903+
+119876Φ+
) gt 0 this implies that119864119862
(2119864 minus 119864119862
) gt 119864119862
(2119864 minus 119864119862
) The corresponding first-orderexpansion of the correction term is
119864119862
(2119864 minus 119864119862
) = 119864119862
(2119864 minus 119864119862
)
+2
1199032+
(119903+
+ 119876Φ+
) (2119864 minus 119864119862
)119872120596
(43)
This shows that the ABGB BH entropy described by CVformula also violates the holographic bound
4 Concluding Remarks
The description of BH entropy by CV formula and itsquantum corrections have attained much attention Therehave been a number of papers to find the semiclassicalcorrections in this formula for different BHs using varioustechniques such as self-gravitational effects [41] generalizeduncertainty principle [42] space noncommutativity [43] andquantum corrections [44] The self-gravitational correctionallows one to study across horizon physics and provides thesolution of information loss paradox
This paper is devoted to study this analysis forthe Bardeen and ABGB BHs which correspond to theSchwarzschild BH for zero charges Firstly we have evaluatedthe thermodynamical quantities such as electric potentialHawking temperature and Bekenstein-Hawking entropy ofboth of the BHs It is found that there exists unique eventhorizon whose entropy can be expressed in the form of CVformula indicating that this formula holds on both eventhorizons
Secondly we have used the self-gravitational effect toevaluate the temperature and entropy of these BHs inthe context of KKW analysis The corrections in the CVformula due to self-gravitational effects are evaluated bytaking dynamical background of the BH The correctionsare restricted up to the linear order in 120596 for the zerothorder the modified temperature corresponds to the Hawkingtemperature We have also found the corrections in all thequantities which can be modified under self-gravitationaleffect that is pure extensive energy and the Casimir energyThe electric potential charge spherical radius and totalenergy remain invariant under this effect It turns out that theself-gravitational correction term in modified CV formula ispositive for both BHs as the modified term is greater than theoriginal It is interesting tomention here that the positive self-gravitational corrections for regular BHs do not satisfy theinequality 119878CFT gt 119878 gt 119878bh which proves that the holographicbound is not universal
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the Higher EducationCommission Islamabad Pakistan for its financial supportthrough the Indigenous PhD Fellowship for 5K ScholarsPhase-II Batch-I
References
[1] J Michell ldquoOn the means of discovering the distance magni-tude ampc of the fixed stars in consequence of the diminution ofthe velocity of their light in case such a diminution should befound to take place in any of them and such other data shouldbe procured from observations as would be farther necessary
6 Advances in High Energy Physics
for that purposerdquoPhilosophical Transactions of the Royal Societyvol 74 pp 35ndash57 1784
[2] J AWheeler ldquoOur universe the known and the unknownrdquoTheAmerican Scientist vol 56 p 1 1968
[3] S W Hawking ldquoBlack hole explosionsrdquo Nature vol 248 no5443 pp 30ndash31 1974
[4] J B Hartle and S W Hawking ldquoPath-integral derivation ofblack-hole radiancerdquo Physical Review D vol 13 no 8 pp 2188ndash2203 1976
[5] Q Q Jiang and X Cai ldquoRemarks on self-interaction correctionto black hole radiationrdquo Journal of High Energy Physics vol2003 no 11 article 110 2009
[6] P Kraus and F Wilczek ldquoSelf-interaction correction to blackhole radiancerdquo Nuclear Physics B vol 433 no 2 pp 403ndash4201995
[7] P Kraus and F Wilczek ldquoEffect of self-interaction on chargedblack hole radiancerdquo Nuclear Physics B vol 437 no 1 pp 231ndash242 1995
[8] P Painleve ldquoLamecanique classique et la theorie de la relativiterdquoComptes Rendus de lrsquoAcademie des Sciences vol 173 pp 677ndash680 1921
[9] E C Vagenas ldquoGeneralization of the KKW analysis for blackhole radiationrdquo Physics Letters B vol 559 no 1-2 pp 65ndash732003
[10] M K Parikh and F Wilczek ldquoHawking radiation as tunnelingrdquoPhysical Review Letters vol 85 no 24 pp 5042ndash5045 2000
[11] S Hemming and E Keski-Vakkuri ldquoHawking radiation fromAdS black holesrdquo Physical Review D vol 64 no 4 Article ID044006 8 pages 2001
[12] Y Kwon ldquoHawking radiation in AdS2
black holerdquo Il NuovoCimento della Societa Italiana di Fisica B Serie 12 vol 115 no 4pp 469ndash471 2000
[13] E C Vagenas ldquoAre extremal 2D black holes really frozenrdquoPhysics Letters B vol 503 no 3-4 pp 399ndash403 2001
[14] E C Vagenas ldquoTwo-dimensional dilatonic black holes andHawking radiationrdquo Modern Physics Letters A Particles andFields Gravitation Cosmology Nuclear Physics vol 17 no 10pp 609ndash618 2002
[15] E C Vagenas ldquoSemiclassical corrections to the Bekenstein-Hawking entropy of the BTZ black hole via self-gravitationrdquoPhysics Letters B vol 533 no 3-4 pp 302ndash306 2002
[16] A J M Medved ldquoRadiation via tunnelling in the charged BTZblack holerdquo Classical and Quantum Gravity vol 19 no 3 pp589ndash598 2002
[17] A J M Medved ldquoRadiation via tunneling from a de Sittercosmological horizonrdquo Physical ReviewD vol 66 no 12 ArticleID 124009 7 pages 2002
[18] L Susskind ldquoTheworld as a hologramrdquo Journal ofMathematicalPhysics vol 36 no 11 pp 6377ndash6396 1995
[19] E Verlinde ldquoOn the holographic principle in a radiationdominateduniverserdquo httparxivorgabshepth0008140
[20] D Birmingham and S Mokhtari ldquoThe Cardy-Verlinde formulaand Taub-bolt-AdS spacetimesrdquo Physics Letters B vol 508 no3-4 pp 365ndash368 2001
[21] D Klemm A C Petkou and G Siopsis ldquoEntropy boundsmonotonicity properties and scaling in CFTsrdquo Nuclear PhysicsB vol 601 no 1-2 pp 380ndash394 2001
[22] B Wang E Abdalla and R-K Su ldquoRelating Friedmannequation to Cardy formula in universes with cosmologicalconstantrdquo Physics Letters B vol 503 no 3-4 pp 394ndash398 2001
[23] M R Setare ldquoThe Cardy-Verlinde formula and entropy oftopological Reissner-Nordstrom black holes in de Sitter spacesrdquoModern Physics Letters A Particles and Fields GravitationCosmology Nuclear Physics vol 17 no 32 pp 2089ndash2094 2002
[24] M R Setare andM B Altaie ldquoTheCardy-Verlinde formula andentropy of topological Kerr-Newman black holes in de Sitterspacesrdquo The European Physical Journal C Particles and Fieldsvol 30 no 2 pp 273ndash277 2003
[25] M R Setare and R Mansouri ldquoHolographic thermodynamicson the brane in topological Reissner-Nordstromde Sitter spacerdquoInternational Journal of Modern Physics A vol 18 no 24 pp4443ndash4450 2003
[26] C O Lee ldquoCardy-Verlinde formula in Taub-NUTbolt-(A)dSspacerdquo Physics Letters B vol 670 no 2 pp 146ndash149 2008
[27] A J M Medved ldquoQuantum-corrected Cardy entropy forgeneric (1 + 1)-dimensional gravityrdquo Classical and QuantumGravity vol 19 no 9 pp 2503ndash2513 2002
[28] SMukherji and S S Pal ldquoLogarithmic corrections to black holeentropy and AdSCFT correspondencerdquo Journal of High EnergyPhysics vol 2002 no 5 article 026 2002
[29] J E Lidsey S Nojiri S D Odintsov and S OgushildquoThe AdSCFT correspondence and logarithmic correctionsto braneworld cosmology and the Cardy-Verlinde formulardquoPhysics Letters B vol 544 no 3-4 pp 337ndash345 2002
[30] S Carlip ldquoLogarithmic corrections to black hole entropy fromthe Cardy formulardquo Classical and Quantum Gravity vol 17 no20 pp 4175ndash4186 2000
[31] M R Setare and E C Vagenas ldquoCardy-Verlinde formula andAchucarro-Ortiz black holerdquo Physical Review D vol 68 no 6Article ID 064014 5 pages 2003
[32] MR Setare andECVagenas ldquoSelf-gravitational corrections tothe Cardy-Verlinde formula of the Achucarro-Ortiz black holerdquoPhysics Letters B vol 584 no 1-2 pp 127ndash132 2004
[33] M R Setare and M Jamil ldquoThe Cardy-Verlinde formula andentropy of the charged rotating BTZ black holerdquo Physics LettersB vol 681 no 5 pp 469ndash471 2009
[34] F Darabi M Jamil and M R Setare ldquoSelf-gravitationalcorrections to theCardy-Verlinde formula of chargedBTZblackholerdquoModern Physics Letters A Particles and Fields GravitationCosmology Nuclear Physics vol 26 no 14 pp 1047ndash1057 2011
[35] G Abbas ldquoCardy-Verlinde formula of noncommutativeSchwarzschild black holerdquo Advances in High Energy Physicsvol 2014 Article ID 306256 4 pages 2014
[36] J Bardeen ldquoNon-singular general relativistic gravitational col-lapserdquo in Proceedings of the 5th International Conference onGravitation and the Theory of Relativity Tbilisi Unversty PressTbilisi Georgia September 1968
[37] E Ayon-Beato and A Garcıa ldquoNew regular black hole solutionfrom nonlinear electrodynamicsrdquo Physics Letters B vol 464 no1-2 pp 25ndash29 1999
[38] K A Bronnikov ldquoComment on lsquoregular black hole in gen-eral relativity coupled to nonlinear electrodynamicsrsquordquo PhysicalReview Letters vol 85 p 4641 2000
[39] R Bousso ldquoA covariant entropy conjecturerdquo Journal of HighEnergy Physics vol 1999 no 7 article 004 1999
[40] G rsquot Hooft ldquoDimensional reduction in quantumgravityrdquohttparxivorgabsgr-qc9310026
[41] M R Setare and E C Vagenas ldquoSelf-gravitational correctionsto the Cardy-Verlinde formula and the FRW brane cosmologyin SdS
5
bulkrdquo International Journal of Modern Physics A vol 20no 30 pp 7219ndash7232 2005
Advances in High Energy Physics 7
[42] M R Setare ldquoThe generalized uncertainty principle and cor-rections to the Cardy-Verlinde formula in SAdS
5
black holesrdquoInternational Journal of Modern Physics A vol 21 no 6 p 13252006
[43] M R Setare ldquoSpace noncommutativity corrections to theCardy-Verlinde formulardquo International Journal of ModernPhysics A vol 21 no 13-14 p 3007 2006
[44] M Sharif and W Javed ldquoQuantum corrections for ABGB blackholerdquoAstrophysics and Space Science vol 337 no 1 pp 335ndash3412012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
![Page 5: Research Article Cardy-Verlinde Formula and Its Self ...downloads.hindawi.com/journals/ahep/2014/926589.pdf · e generalized Cardy-Verlinde formula is de ned as [ ] CFT = 2 % & '*](https://reader035.vdocuments.net/reader035/viewer/2022071219/6056d064e08cc116434a5561/html5/thumbnails/5.jpg)
Advances in High Energy Physics 5
32 Self-Gravitational Corrections for Ayon Beato-Garcıa-Bronnikov Black Hole Now we consider ABGB BH tocalculate the self-gravitational corrections in the CV formulausing KKW analysis The corresponding self-gravitationalcorrections in 119879bh119878bh are
119879bh119878bh = 119879119867119878 minus120596
2119872(119903+
+ 119876Φ+
) (39)
Equation (29) gives the modified Casimir energy as
119864119862
=3
2119872 +
5
4(1198762
119903+
sec ℎ2 ( 1198762
2119898119903+
)) +2
1199032+
(119903+
+ 119876Φ+
)119872120596
= 119864119862
+2
1199032+
(119903+
+ 119876Φ+
)119872120596
(40)
where the second term is the correction term Additionallythe modified extensive energy can be written as
2119864 minus 119864119862
= 2119864 minus 119864119862
minus2
1199032+
(119903+
+ 119876Φ+
)119872120596
119864119864
= 119864119864
minus1
1199032+
(119903+
+ 119876Φ+
)119872120596
(41)
Inserting (27) (40) and (41) in modified CV formula weobtain the self-gravitational corrected CV formula for ABGBBH as follows
119878CFT
= 119878[
[
((3
8119872119903+
minus4
1199032+
(119903+
+ 119876Φ+
)1198722
120596
minus25
16
1198764
1199032+
sec ℎ4 ( 1198762
2119898119903+
))
times ((1
8119903+
minus5
8
1198764
1199032+
sec ℎ4 ( 1198762
2119898119903+
))
times (3
2119872 +
5
4
1198764
1199032+
sec ℎ4 ( 1198762
2119898119903+
)))
minus1
)
12
]
]
(42)
As the correction term (21199032
+
)(119903+
+119876Φ+
) gt 0 this implies that119864119862
(2119864 minus 119864119862
) gt 119864119862
(2119864 minus 119864119862
) The corresponding first-orderexpansion of the correction term is
119864119862
(2119864 minus 119864119862
) = 119864119862
(2119864 minus 119864119862
)
+2
1199032+
(119903+
+ 119876Φ+
) (2119864 minus 119864119862
)119872120596
(43)
This shows that the ABGB BH entropy described by CVformula also violates the holographic bound
4 Concluding Remarks
The description of BH entropy by CV formula and itsquantum corrections have attained much attention Therehave been a number of papers to find the semiclassicalcorrections in this formula for different BHs using varioustechniques such as self-gravitational effects [41] generalizeduncertainty principle [42] space noncommutativity [43] andquantum corrections [44] The self-gravitational correctionallows one to study across horizon physics and provides thesolution of information loss paradox
This paper is devoted to study this analysis forthe Bardeen and ABGB BHs which correspond to theSchwarzschild BH for zero charges Firstly we have evaluatedthe thermodynamical quantities such as electric potentialHawking temperature and Bekenstein-Hawking entropy ofboth of the BHs It is found that there exists unique eventhorizon whose entropy can be expressed in the form of CVformula indicating that this formula holds on both eventhorizons
Secondly we have used the self-gravitational effect toevaluate the temperature and entropy of these BHs inthe context of KKW analysis The corrections in the CVformula due to self-gravitational effects are evaluated bytaking dynamical background of the BH The correctionsare restricted up to the linear order in 120596 for the zerothorder the modified temperature corresponds to the Hawkingtemperature We have also found the corrections in all thequantities which can be modified under self-gravitationaleffect that is pure extensive energy and the Casimir energyThe electric potential charge spherical radius and totalenergy remain invariant under this effect It turns out that theself-gravitational correction term in modified CV formula ispositive for both BHs as the modified term is greater than theoriginal It is interesting tomention here that the positive self-gravitational corrections for regular BHs do not satisfy theinequality 119878CFT gt 119878 gt 119878bh which proves that the holographicbound is not universal
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the Higher EducationCommission Islamabad Pakistan for its financial supportthrough the Indigenous PhD Fellowship for 5K ScholarsPhase-II Batch-I
References
[1] J Michell ldquoOn the means of discovering the distance magni-tude ampc of the fixed stars in consequence of the diminution ofthe velocity of their light in case such a diminution should befound to take place in any of them and such other data shouldbe procured from observations as would be farther necessary
6 Advances in High Energy Physics
for that purposerdquoPhilosophical Transactions of the Royal Societyvol 74 pp 35ndash57 1784
[2] J AWheeler ldquoOur universe the known and the unknownrdquoTheAmerican Scientist vol 56 p 1 1968
[3] S W Hawking ldquoBlack hole explosionsrdquo Nature vol 248 no5443 pp 30ndash31 1974
[4] J B Hartle and S W Hawking ldquoPath-integral derivation ofblack-hole radiancerdquo Physical Review D vol 13 no 8 pp 2188ndash2203 1976
[5] Q Q Jiang and X Cai ldquoRemarks on self-interaction correctionto black hole radiationrdquo Journal of High Energy Physics vol2003 no 11 article 110 2009
[6] P Kraus and F Wilczek ldquoSelf-interaction correction to blackhole radiancerdquo Nuclear Physics B vol 433 no 2 pp 403ndash4201995
[7] P Kraus and F Wilczek ldquoEffect of self-interaction on chargedblack hole radiancerdquo Nuclear Physics B vol 437 no 1 pp 231ndash242 1995
[8] P Painleve ldquoLamecanique classique et la theorie de la relativiterdquoComptes Rendus de lrsquoAcademie des Sciences vol 173 pp 677ndash680 1921
[9] E C Vagenas ldquoGeneralization of the KKW analysis for blackhole radiationrdquo Physics Letters B vol 559 no 1-2 pp 65ndash732003
[10] M K Parikh and F Wilczek ldquoHawking radiation as tunnelingrdquoPhysical Review Letters vol 85 no 24 pp 5042ndash5045 2000
[11] S Hemming and E Keski-Vakkuri ldquoHawking radiation fromAdS black holesrdquo Physical Review D vol 64 no 4 Article ID044006 8 pages 2001
[12] Y Kwon ldquoHawking radiation in AdS2
black holerdquo Il NuovoCimento della Societa Italiana di Fisica B Serie 12 vol 115 no 4pp 469ndash471 2000
[13] E C Vagenas ldquoAre extremal 2D black holes really frozenrdquoPhysics Letters B vol 503 no 3-4 pp 399ndash403 2001
[14] E C Vagenas ldquoTwo-dimensional dilatonic black holes andHawking radiationrdquo Modern Physics Letters A Particles andFields Gravitation Cosmology Nuclear Physics vol 17 no 10pp 609ndash618 2002
[15] E C Vagenas ldquoSemiclassical corrections to the Bekenstein-Hawking entropy of the BTZ black hole via self-gravitationrdquoPhysics Letters B vol 533 no 3-4 pp 302ndash306 2002
[16] A J M Medved ldquoRadiation via tunnelling in the charged BTZblack holerdquo Classical and Quantum Gravity vol 19 no 3 pp589ndash598 2002
[17] A J M Medved ldquoRadiation via tunneling from a de Sittercosmological horizonrdquo Physical ReviewD vol 66 no 12 ArticleID 124009 7 pages 2002
[18] L Susskind ldquoTheworld as a hologramrdquo Journal ofMathematicalPhysics vol 36 no 11 pp 6377ndash6396 1995
[19] E Verlinde ldquoOn the holographic principle in a radiationdominateduniverserdquo httparxivorgabshepth0008140
[20] D Birmingham and S Mokhtari ldquoThe Cardy-Verlinde formulaand Taub-bolt-AdS spacetimesrdquo Physics Letters B vol 508 no3-4 pp 365ndash368 2001
[21] D Klemm A C Petkou and G Siopsis ldquoEntropy boundsmonotonicity properties and scaling in CFTsrdquo Nuclear PhysicsB vol 601 no 1-2 pp 380ndash394 2001
[22] B Wang E Abdalla and R-K Su ldquoRelating Friedmannequation to Cardy formula in universes with cosmologicalconstantrdquo Physics Letters B vol 503 no 3-4 pp 394ndash398 2001
[23] M R Setare ldquoThe Cardy-Verlinde formula and entropy oftopological Reissner-Nordstrom black holes in de Sitter spacesrdquoModern Physics Letters A Particles and Fields GravitationCosmology Nuclear Physics vol 17 no 32 pp 2089ndash2094 2002
[24] M R Setare andM B Altaie ldquoTheCardy-Verlinde formula andentropy of topological Kerr-Newman black holes in de Sitterspacesrdquo The European Physical Journal C Particles and Fieldsvol 30 no 2 pp 273ndash277 2003
[25] M R Setare and R Mansouri ldquoHolographic thermodynamicson the brane in topological Reissner-Nordstromde Sitter spacerdquoInternational Journal of Modern Physics A vol 18 no 24 pp4443ndash4450 2003
[26] C O Lee ldquoCardy-Verlinde formula in Taub-NUTbolt-(A)dSspacerdquo Physics Letters B vol 670 no 2 pp 146ndash149 2008
[27] A J M Medved ldquoQuantum-corrected Cardy entropy forgeneric (1 + 1)-dimensional gravityrdquo Classical and QuantumGravity vol 19 no 9 pp 2503ndash2513 2002
[28] SMukherji and S S Pal ldquoLogarithmic corrections to black holeentropy and AdSCFT correspondencerdquo Journal of High EnergyPhysics vol 2002 no 5 article 026 2002
[29] J E Lidsey S Nojiri S D Odintsov and S OgushildquoThe AdSCFT correspondence and logarithmic correctionsto braneworld cosmology and the Cardy-Verlinde formulardquoPhysics Letters B vol 544 no 3-4 pp 337ndash345 2002
[30] S Carlip ldquoLogarithmic corrections to black hole entropy fromthe Cardy formulardquo Classical and Quantum Gravity vol 17 no20 pp 4175ndash4186 2000
[31] M R Setare and E C Vagenas ldquoCardy-Verlinde formula andAchucarro-Ortiz black holerdquo Physical Review D vol 68 no 6Article ID 064014 5 pages 2003
[32] MR Setare andECVagenas ldquoSelf-gravitational corrections tothe Cardy-Verlinde formula of the Achucarro-Ortiz black holerdquoPhysics Letters B vol 584 no 1-2 pp 127ndash132 2004
[33] M R Setare and M Jamil ldquoThe Cardy-Verlinde formula andentropy of the charged rotating BTZ black holerdquo Physics LettersB vol 681 no 5 pp 469ndash471 2009
[34] F Darabi M Jamil and M R Setare ldquoSelf-gravitationalcorrections to theCardy-Verlinde formula of chargedBTZblackholerdquoModern Physics Letters A Particles and Fields GravitationCosmology Nuclear Physics vol 26 no 14 pp 1047ndash1057 2011
[35] G Abbas ldquoCardy-Verlinde formula of noncommutativeSchwarzschild black holerdquo Advances in High Energy Physicsvol 2014 Article ID 306256 4 pages 2014
[36] J Bardeen ldquoNon-singular general relativistic gravitational col-lapserdquo in Proceedings of the 5th International Conference onGravitation and the Theory of Relativity Tbilisi Unversty PressTbilisi Georgia September 1968
[37] E Ayon-Beato and A Garcıa ldquoNew regular black hole solutionfrom nonlinear electrodynamicsrdquo Physics Letters B vol 464 no1-2 pp 25ndash29 1999
[38] K A Bronnikov ldquoComment on lsquoregular black hole in gen-eral relativity coupled to nonlinear electrodynamicsrsquordquo PhysicalReview Letters vol 85 p 4641 2000
[39] R Bousso ldquoA covariant entropy conjecturerdquo Journal of HighEnergy Physics vol 1999 no 7 article 004 1999
[40] G rsquot Hooft ldquoDimensional reduction in quantumgravityrdquohttparxivorgabsgr-qc9310026
[41] M R Setare and E C Vagenas ldquoSelf-gravitational correctionsto the Cardy-Verlinde formula and the FRW brane cosmologyin SdS
5
bulkrdquo International Journal of Modern Physics A vol 20no 30 pp 7219ndash7232 2005
Advances in High Energy Physics 7
[42] M R Setare ldquoThe generalized uncertainty principle and cor-rections to the Cardy-Verlinde formula in SAdS
5
black holesrdquoInternational Journal of Modern Physics A vol 21 no 6 p 13252006
[43] M R Setare ldquoSpace noncommutativity corrections to theCardy-Verlinde formulardquo International Journal of ModernPhysics A vol 21 no 13-14 p 3007 2006
[44] M Sharif and W Javed ldquoQuantum corrections for ABGB blackholerdquoAstrophysics and Space Science vol 337 no 1 pp 335ndash3412012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
![Page 6: Research Article Cardy-Verlinde Formula and Its Self ...downloads.hindawi.com/journals/ahep/2014/926589.pdf · e generalized Cardy-Verlinde formula is de ned as [ ] CFT = 2 % & '*](https://reader035.vdocuments.net/reader035/viewer/2022071219/6056d064e08cc116434a5561/html5/thumbnails/6.jpg)
6 Advances in High Energy Physics
for that purposerdquoPhilosophical Transactions of the Royal Societyvol 74 pp 35ndash57 1784
[2] J AWheeler ldquoOur universe the known and the unknownrdquoTheAmerican Scientist vol 56 p 1 1968
[3] S W Hawking ldquoBlack hole explosionsrdquo Nature vol 248 no5443 pp 30ndash31 1974
[4] J B Hartle and S W Hawking ldquoPath-integral derivation ofblack-hole radiancerdquo Physical Review D vol 13 no 8 pp 2188ndash2203 1976
[5] Q Q Jiang and X Cai ldquoRemarks on self-interaction correctionto black hole radiationrdquo Journal of High Energy Physics vol2003 no 11 article 110 2009
[6] P Kraus and F Wilczek ldquoSelf-interaction correction to blackhole radiancerdquo Nuclear Physics B vol 433 no 2 pp 403ndash4201995
[7] P Kraus and F Wilczek ldquoEffect of self-interaction on chargedblack hole radiancerdquo Nuclear Physics B vol 437 no 1 pp 231ndash242 1995
[8] P Painleve ldquoLamecanique classique et la theorie de la relativiterdquoComptes Rendus de lrsquoAcademie des Sciences vol 173 pp 677ndash680 1921
[9] E C Vagenas ldquoGeneralization of the KKW analysis for blackhole radiationrdquo Physics Letters B vol 559 no 1-2 pp 65ndash732003
[10] M K Parikh and F Wilczek ldquoHawking radiation as tunnelingrdquoPhysical Review Letters vol 85 no 24 pp 5042ndash5045 2000
[11] S Hemming and E Keski-Vakkuri ldquoHawking radiation fromAdS black holesrdquo Physical Review D vol 64 no 4 Article ID044006 8 pages 2001
[12] Y Kwon ldquoHawking radiation in AdS2
black holerdquo Il NuovoCimento della Societa Italiana di Fisica B Serie 12 vol 115 no 4pp 469ndash471 2000
[13] E C Vagenas ldquoAre extremal 2D black holes really frozenrdquoPhysics Letters B vol 503 no 3-4 pp 399ndash403 2001
[14] E C Vagenas ldquoTwo-dimensional dilatonic black holes andHawking radiationrdquo Modern Physics Letters A Particles andFields Gravitation Cosmology Nuclear Physics vol 17 no 10pp 609ndash618 2002
[15] E C Vagenas ldquoSemiclassical corrections to the Bekenstein-Hawking entropy of the BTZ black hole via self-gravitationrdquoPhysics Letters B vol 533 no 3-4 pp 302ndash306 2002
[16] A J M Medved ldquoRadiation via tunnelling in the charged BTZblack holerdquo Classical and Quantum Gravity vol 19 no 3 pp589ndash598 2002
[17] A J M Medved ldquoRadiation via tunneling from a de Sittercosmological horizonrdquo Physical ReviewD vol 66 no 12 ArticleID 124009 7 pages 2002
[18] L Susskind ldquoTheworld as a hologramrdquo Journal ofMathematicalPhysics vol 36 no 11 pp 6377ndash6396 1995
[19] E Verlinde ldquoOn the holographic principle in a radiationdominateduniverserdquo httparxivorgabshepth0008140
[20] D Birmingham and S Mokhtari ldquoThe Cardy-Verlinde formulaand Taub-bolt-AdS spacetimesrdquo Physics Letters B vol 508 no3-4 pp 365ndash368 2001
[21] D Klemm A C Petkou and G Siopsis ldquoEntropy boundsmonotonicity properties and scaling in CFTsrdquo Nuclear PhysicsB vol 601 no 1-2 pp 380ndash394 2001
[22] B Wang E Abdalla and R-K Su ldquoRelating Friedmannequation to Cardy formula in universes with cosmologicalconstantrdquo Physics Letters B vol 503 no 3-4 pp 394ndash398 2001
[23] M R Setare ldquoThe Cardy-Verlinde formula and entropy oftopological Reissner-Nordstrom black holes in de Sitter spacesrdquoModern Physics Letters A Particles and Fields GravitationCosmology Nuclear Physics vol 17 no 32 pp 2089ndash2094 2002
[24] M R Setare andM B Altaie ldquoTheCardy-Verlinde formula andentropy of topological Kerr-Newman black holes in de Sitterspacesrdquo The European Physical Journal C Particles and Fieldsvol 30 no 2 pp 273ndash277 2003
[25] M R Setare and R Mansouri ldquoHolographic thermodynamicson the brane in topological Reissner-Nordstromde Sitter spacerdquoInternational Journal of Modern Physics A vol 18 no 24 pp4443ndash4450 2003
[26] C O Lee ldquoCardy-Verlinde formula in Taub-NUTbolt-(A)dSspacerdquo Physics Letters B vol 670 no 2 pp 146ndash149 2008
[27] A J M Medved ldquoQuantum-corrected Cardy entropy forgeneric (1 + 1)-dimensional gravityrdquo Classical and QuantumGravity vol 19 no 9 pp 2503ndash2513 2002
[28] SMukherji and S S Pal ldquoLogarithmic corrections to black holeentropy and AdSCFT correspondencerdquo Journal of High EnergyPhysics vol 2002 no 5 article 026 2002
[29] J E Lidsey S Nojiri S D Odintsov and S OgushildquoThe AdSCFT correspondence and logarithmic correctionsto braneworld cosmology and the Cardy-Verlinde formulardquoPhysics Letters B vol 544 no 3-4 pp 337ndash345 2002
[30] S Carlip ldquoLogarithmic corrections to black hole entropy fromthe Cardy formulardquo Classical and Quantum Gravity vol 17 no20 pp 4175ndash4186 2000
[31] M R Setare and E C Vagenas ldquoCardy-Verlinde formula andAchucarro-Ortiz black holerdquo Physical Review D vol 68 no 6Article ID 064014 5 pages 2003
[32] MR Setare andECVagenas ldquoSelf-gravitational corrections tothe Cardy-Verlinde formula of the Achucarro-Ortiz black holerdquoPhysics Letters B vol 584 no 1-2 pp 127ndash132 2004
[33] M R Setare and M Jamil ldquoThe Cardy-Verlinde formula andentropy of the charged rotating BTZ black holerdquo Physics LettersB vol 681 no 5 pp 469ndash471 2009
[34] F Darabi M Jamil and M R Setare ldquoSelf-gravitationalcorrections to theCardy-Verlinde formula of chargedBTZblackholerdquoModern Physics Letters A Particles and Fields GravitationCosmology Nuclear Physics vol 26 no 14 pp 1047ndash1057 2011
[35] G Abbas ldquoCardy-Verlinde formula of noncommutativeSchwarzschild black holerdquo Advances in High Energy Physicsvol 2014 Article ID 306256 4 pages 2014
[36] J Bardeen ldquoNon-singular general relativistic gravitational col-lapserdquo in Proceedings of the 5th International Conference onGravitation and the Theory of Relativity Tbilisi Unversty PressTbilisi Georgia September 1968
[37] E Ayon-Beato and A Garcıa ldquoNew regular black hole solutionfrom nonlinear electrodynamicsrdquo Physics Letters B vol 464 no1-2 pp 25ndash29 1999
[38] K A Bronnikov ldquoComment on lsquoregular black hole in gen-eral relativity coupled to nonlinear electrodynamicsrsquordquo PhysicalReview Letters vol 85 p 4641 2000
[39] R Bousso ldquoA covariant entropy conjecturerdquo Journal of HighEnergy Physics vol 1999 no 7 article 004 1999
[40] G rsquot Hooft ldquoDimensional reduction in quantumgravityrdquohttparxivorgabsgr-qc9310026
[41] M R Setare and E C Vagenas ldquoSelf-gravitational correctionsto the Cardy-Verlinde formula and the FRW brane cosmologyin SdS
5
bulkrdquo International Journal of Modern Physics A vol 20no 30 pp 7219ndash7232 2005
Advances in High Energy Physics 7
[42] M R Setare ldquoThe generalized uncertainty principle and cor-rections to the Cardy-Verlinde formula in SAdS
5
black holesrdquoInternational Journal of Modern Physics A vol 21 no 6 p 13252006
[43] M R Setare ldquoSpace noncommutativity corrections to theCardy-Verlinde formulardquo International Journal of ModernPhysics A vol 21 no 13-14 p 3007 2006
[44] M Sharif and W Javed ldquoQuantum corrections for ABGB blackholerdquoAstrophysics and Space Science vol 337 no 1 pp 335ndash3412012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
![Page 7: Research Article Cardy-Verlinde Formula and Its Self ...downloads.hindawi.com/journals/ahep/2014/926589.pdf · e generalized Cardy-Verlinde formula is de ned as [ ] CFT = 2 % & '*](https://reader035.vdocuments.net/reader035/viewer/2022071219/6056d064e08cc116434a5561/html5/thumbnails/7.jpg)
Advances in High Energy Physics 7
[42] M R Setare ldquoThe generalized uncertainty principle and cor-rections to the Cardy-Verlinde formula in SAdS
5
black holesrdquoInternational Journal of Modern Physics A vol 21 no 6 p 13252006
[43] M R Setare ldquoSpace noncommutativity corrections to theCardy-Verlinde formulardquo International Journal of ModernPhysics A vol 21 no 13-14 p 3007 2006
[44] M Sharif and W Javed ldquoQuantum corrections for ABGB blackholerdquoAstrophysics and Space Science vol 337 no 1 pp 335ndash3412012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
![Page 8: Research Article Cardy-Verlinde Formula and Its Self ...downloads.hindawi.com/journals/ahep/2014/926589.pdf · e generalized Cardy-Verlinde formula is de ned as [ ] CFT = 2 % & '*](https://reader035.vdocuments.net/reader035/viewer/2022071219/6056d064e08cc116434a5561/html5/thumbnails/8.jpg)
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of