Research ArticleQuantum Tunnelling for Hawking Radiation fromBoth Static and Dynamic Black Holes

Subenoy Chakraborty and Subhajit Saha

Department of Mathematics Jadavpur University Kolkata West Bengal 700032 India

Correspondence should be addressed to Subenoy Chakraborty schakrabortymathgmailcom

Received 31 December 2013 Revised 12 March 2014 Accepted 25 March 2014 Published 23 April 2014

Academic Editor Christian Corda

Copyright copy 2014 S Chakraborty and S Saha 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

The paper deals with Hawking radiation from both a general static black hole and a nonstatic spherically symmetric black holeIn case of static black hole tunnelling of nonzero mass particles is considered and due to complicated calculations quantumcorrections are calculated only up to the first order The results are compared with those for massless particles near the horizonOn the other hand for dynamical black hole quantum corrections are incorporated using the Hamilton-Jacobi method beyondsemiclassical approximation It is found that different order correction terms satisfy identical differential equation and are solvedby a typical technique Finally using the law of black hole mechanics a general modified form of the black hole entropy is obtainedconsidering modified Hawking temperature

1 Introduction

Hawking radiation is one of the most important effects inblack hole (BH) physics Classically nothing can escape fromthe BH across its event horizon But in 1974 there was adramatic change in view when Hawking and Hartle [1 2]showed that BHs are not totally black they radiate analogousto thermal black body radiation Since then there has beenlots of attraction to this issue and various approaches havebeen developed to derive Hawking radiation and its corre-sponding temperature [3ndash7] However in the last decade twodistinct semiclassical methods have been developed whichenhanced the study of Hawking radiation to a great extentThe first approach developed by Parikh and Wilczek [8 9]is based on the heuristic pictures of visualisation of thesource of radiation as tunnelling and is known as radial nullgeodesic method The essence of this method is to calculatethe imaginary part of the action for the s-wave emission(across the horizon) using the radial null geodesic equationand is then related to the Boltzmann factor to obtainHawkingradiation by the relation

Γ prop exp minus2ℎ(Im 119878

outminus Im 119878

in) = expminus 119864

119879119867

(1)

where 119864 is the energy associated with the tunnelling particleand 119879

119867is the usual Hawking temperature

The alternative way of looking into this aspect is known ascomplex paths method developed by Srinivasan et al [10 11]In this approach the differential equation of the action 119878(119903 119905)of a classical scalar particle can be obtained by plugging thescalar field wave function 120601(119903 119905) = expminus(119894ℏ)119878(119903 119905) into theKlein-Gordon (KG) equation in a gravitational backgroundThen the Hamilton-Jacobi (HJ) method is employed to solvethe differential equation for 119878 Finally Hawking temperatureis obtained using the ldquoprinciple of detailed balancerdquo [10ndash12](time-reversal invariant) It should be noted that the firstmethod is limited tomassless particles only Also thismethodis applicable to such coordinate system only in which thereis no singularity across the horizon On the other hand incomplex paths method the emitted particles are consideredwithout self-gravitation and the action is assumed to satisfythe relativistic HJ equation Here tunnelling of both masslessand massive particles is possible and it is applicable to anycoordinate system to describe the BH

Most of the studies [13ndash18] dealing with the Hawkingradiation are connected to semiclassical analysis RecentlyBanerjee and Majhi [19] and Corda et al [20 21] initiated

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2014 Article ID 168487 9 pageshttpdxdoiorg1011552014168487

2 Advances in High Energy Physics

the calculation ofHawking temperature beyond the semiclas-sical limit Mostly both groups have considered tunnellingof massless particle and evaluated the modified Hawkingtemperature with quantum corrections

In the present work at first we consider a generalnonstatic metric for dynamical BH HJ method is extendedbeyond semiclassical approximation to consider all the termsin the expansion of the one particle action It is foundthat the higher order terms (quantum corrections) satisfyidentical differential equations as the semiclassical actionand the complicated terms are eliminated considering BHhorizon as one way barrier We derive the modified Hawkingtemperature using both the above approaches which arefound to be identical at the semiclassical level Also modifiedform of the BH entropy with quantum correction has beenevaluated

Subsequently in the next section we consider tun-nelling of particles having nonzeromass beyond semiclassicalapproximation Due to nonzero mass the imaginary partof the action cannot be evaluated using first approach onlyHJ method will be applicable Further the complicated formof the equations involved restricted us to only first orderquantum correction

2 Method of Radial Null GeodesicA Survey of Earlier Works

This section deals with a brief survey of the method of radialnull geodesicsmethod [8] considering the picture ofHawkingradiation as quantum tunnelling In a word the methodcorrelates the imaginary part of the action for the classicallyforbidden process of s-wave emission across the horizonwith the Boltzmann factor for the black body radiation atthe Hawking temperature We start with a general class ofnonstatic spherically symmetric BH metric of the form

1198891199042= minus119860 (119903 119905) 119889119905

2+

1198891199032

119861 (119903 119905)+ 1199032119889Ω2

2 (2)

where the horizon 119903ℎis located at 119860(119903

ℎ 119905) = 0 = 119861(119903

ℎ 119905)

and the metric has a coordinate singularity at the horizonTo remove the coordinate singularity we make the followingPainleve-type transformation of coordinates

119889119905 997888rarr 119889119905 minus radic1 minus 119861

119860119861119889119903 (3)

and as a result metric (2) transforms to

1198891199042= minus119860119889119905

2+ 2radic119860(

1

119861minus 1)119889119905119889119903 + 119889119903

2+ 1199032119889Ω2

2 (4)

This metric (ie the choice of coordinates) has some distinctfeatures over the former one as follows

(i) The metric is singularity free across the horizon(ii) At any fixed time we have a flat spatial geometry(iii) Both the metric will have the same boundary geome-

try at any fixed radius

The radial null geodesic (characterized by 1198891199042 = 0 =

119889Ω2

2) has the differential equation (using (3))

119889119903

119889119905= radic

119860

119861[plusmn1 minus radic1 minus 119861 (119903 119905)] (5)

where outgoing or ingoing geodesic is identified by the + orminus sign within the square bracket in (4) In the present casewe deal with the absorption of particles through the horizon(ie + sign only) and according to Parikh and Wilczek [8]the imaginary part of the action is obtained as

Im 119878 = Imint

119903out

119903in

119901119903119889119903 = Imint

119903out

119903in

int

119901119903

0

1198891199011015840

119903119889119903

= Imint

119903out

119903in

int

119867

0

1198891198671015840

119889119903119889119905 119889119903

(6)

Note that in the last step of the above derivation we haveused the Hamiltonrsquos equation 119903 = (119889119867119889119901

119903)|119903 where (119903119901

119903)

are canonical pair Further it is to be mentioned that inquantum mechanics the action of a tunnelled particle in apotential barrier having energy larger than the energy of theparticle will be imaginary as 119901

119903= radic2119898(119864 minus 119881) For the

present nonstatic BH the mass of the BH is not constantand hence the 1198891198671015840 integration extends over all the values ofenergy of outgoing particle fromzero to119864(119905) [22] (say) Asweare dealing with tunnelling across the BH horizon so usingTaylor series expansion about the horizon 119903

ℎwe write

119860(119903 119905)|119905 =120597119860(119903 119905)

120597119903

10038161003816100381610038161003816100381610038161003816119905

(119903 minus 119903ℎ) + 119874(119903 minus 119903

ℎ)210038161003816100381610038161003816119905

119861(119903 119905)|119905 =120597119861(119903 119905)

120597119903

10038161003816100381610038161003816100381610038161003816119905

(119903 minus 119903ℎ) + 119874(119903 minus 119903

ℎ)210038161003816100381610038161003816119905

(7)

So in the neighbourhood of the horizon the geodesicequation (4) can be approximated as

119889119903

119889119905asymp1

2radic1198601015840 (119903

ℎ 119905) 1198611015840 (119903

ℎ 119905) (119903 minus 119903

ℎ) (8)

Substituting this value of 119889119903119889119905 in the last step of (5) we have

Im 119878 =2120587119864 (119905)

radic1198601015840 (119903ℎ 119905) 1198611015840 (119903

ℎ 119905)

(9)

where the choice of contour for 119903-integration is on the upperhalf complex plane to avoid the coordinate singularity at 119903

ℎ

Thus the tunnelling probability is given by

Γ sim exp minus2ℏIm 119878 = expminus 4120587119864 (119905)

ℏradic11986010158401198611015840 (10)

which in turn equateswith the Boltzmann factor exp119864(119905)119879the expression for the Hawking temperature is

119879119867=

ℏradic1198601015840 (119903ℎ 119905) 1198611015840 (119903

ℎ 119905)

4120587

(11)

Advances in High Energy Physics 3

From the above expression it is to be noted that 119879119867is time

dependentRecently a drawback of the above approach

has been noted [23ndash25] It has been shown thatΓ sim expminus(2ℏ) Im 119878 = expminus(2ℏ) Imint

119903out

119903in119901119903119889119903 is not

canonically invariant and hence is not a proper observable itshould be modified as expminus Im∮119901

119903119889119903ℏ The closed path

goes across the horizon and back For tunnelling across theordinary barrier it is immaterial whether the particle goesfrom the left to the right or the reverse path So in that case

∮119901119903119889119903 = 2int

119903out

119903in

119901119903119889119903 (12)

and there is no problem of canonical invariance But difficultyarises for BH horizon which behaves as a barrier for particlesgoing from inside of the BH to outside but it does not actas a barrier for particles going from outside to the insideSo relation (12) is no longer valid Also using tunnelling theprobability is Γ sim expminus Im∮119901

119903119889119903ℏ so there will be a

problem of factor two in Hawking temperature [24 26 27]Further the above analysis of tunnelling approach

remains incomplete unless effects of self-gravitation and backreaction are taken into account But unfortunately no generalapproaches to account for the above effects are there in theliterature only few results are available for some known BHsolutions [26ndash32]

Finally it is worth mentioning that so far the abovetunnelling approach is purely semiclassical in nature andquantum corrections are not included Also this method isapplicable for Painleve-type coordinates only one cannot usethe original metric coordinates to avoid horizon singularityLastly the tunnelling approach is not applicable for massiveparticles [19]

3 Hamilton-Jacobi MethodQuantum Corrections

We will now follow the alternative approach as mentioned inthe introduction that is the HJ method to evaluate the imag-inary part of the action and hence the Hawking temperatureWe will analyze the beyond semiclassical approximation byincorporating possible quantum corrections As this methodis not affected by the coordinate singularity at the horizon sowe will use the general BH metric (2) for convenience

In the background of the gravitational field describedby the metric (2) massless scalar particles obey the Klein-Gordon equation

ℏ2

radicminus119892120597 [119892120583]radicminus119892120597]] 120595 = 0 (13)

For spherically symmetric BH as we are only consideringradial trajectories so we will consider (119905 119903)-sector in thespacetime given by (2) that is we concentrate on two-dimensional BH problems Using (2) the above Klein-Gordon equation becomes

1205972120595

1205971199052minus

1

2119860119861

120597 (119860119861)

120597119905

120597120595

120597119905minus1

2

120597 (119860119861)

120597119903

120597120595

120597119903minus 119860119861

1205972120595

1205971199032= 0 (14)

Using the standard ansatz for the semiclassical wave functionnamely

120595 (119903 119905) = exp minus 119894ℏ119878 (119903 119905) (15)

the differential equation for the action 119878 is

(120597119878

120597119905)

2

minus 119860119861(120597119878

120597119903)

2

+ 119894ℏ [1205972119878

1205971199052minus

1

2119860119861

120597 (119860119861)

120597119905

120597119878

120597119905minus1

2

120597 (119860119861)

120597119903

120597119878

120597119903minus 119860119861

1205972119878

1205971199032]

(16)

To solve this partial differential equation we expand theaction 119878 in powers of Planckrsquos constant ℏ as

119878 (119903 119905) = 1198780 (119903 119905) + Σℏ119896119878119896 (119903 119905) (17)

with 119896 being a positive integer Note that in the aboveexpansion terms of the order of Planckrsquos constant and itshigher powers are considered as quantum corrections overthe semiclassical action 119878

0 Now substituting ansatz (17) for 119878

into (16) and equating different powers of ℏ on both sides weobtain the following set of partial differential equations

ℏ0 (120597119878

120597119905)

2

minus 119860119861(120597119878

120597119903)

2

= 0 (18)

ℏ11205971198780

120597119905

1205971198781

120597119905minus 119860119861

1205971198780

120597119903

1205971198781

120597119903

+119894

2[12059721198780

1205971199052minus

1

2119860119861

120597 (119860119861)

120597119905

1205971198780

120597119905

minus1

2

120597 (119860119861)

120597119903

1205971198780

120597119903minus 119860119861

12059721198780

1205971199032] = 0

(19)

ℏ2 (1205971198781

120597119905)

2

+ 21205971198780

120597119905

1205971198782

120597119905minus 119860119861(

1205971198781

120597119903)

2

minus 21198601198611205971198780

120597119903

1205971198782

120597119903

+ 119894 [12059721198781

1205971199052minus

1

2119860119861

120597 (119860119861)

120597119905

1205971198781

120597119905

minus1

2

120597 (119860119861)

120597119903

1205971198781

120597119903minus 119860119861

12059721198781

1205971199032] = 0

(20)

and so onApparently different order partial differential equations

are very complicated but fortunately there will be lot ofsimplifications if in the partial differential equation corre-sponding to ℏ119896 all previous partial differential equationsare used and finally we obtain identical partial differentialequation namely

ℏ119896120597119878119896

120597119905= plusmnradic119860 (119903 119905) 119861 (119903 119905)

120597119878119896

120597119903 (21)

for 119896 = 0 1 2

4 Advances in High Energy Physics

Thus quantum corrections satisfy the same differentialequation as the semiclassical action 119878

0 Hence the solutions

will be very similar To solve 1198780 it is to be noted that due to

nonstatic BHs the metric coefficients are functions of 119903 and119905 and hence standard HJ method cannot be applied somegeneralization is needed We start with a general metric [22]

1198780 (119903 119905) = int

119905

0

1205960(1199051015840) 119889119905 + 119863

0 (119903 119905) (22)

Here 1205960(119905) behaves as the energy of the emitted particle

and the justification of the choice of the integral is that theoutgoing particle should have time-dependent continuumenergy

Now substituting the above ansatz for 1198780(119903 119905) into (18) and

using the radial null geodesic in the usual metric from (2)namely

119889119903

119889119905= plusmnradic119860119861 (23)

we have

1205971198630

120597119903+1205971198630

120597119905

119889119905

119889119903= ∓1205960 (119905)

119889119905

119889119903 (24)

that is

1198891198630

119889119903= ∓

1205960 (119905)

radic119860119861

(25)

which gives

1198630= ∓1205960 (119905) int

119903

0

119889119903

radic119860119861

(26)

Hence the complete semiclassical action takes the form

1198780 (119903 119905) = int

119905

0

1205960(1199051015840) 1198891199051015840∓ 1205960 (119905) int

119903

0

119889119903

radic119860119861

(27)

Here the minus (or +) sign corresponds to absorption (or emis-sion) particle As solution (27) contains an arbitrary time-dependent function 120596

0(119905) so a general solution for 119878

119896can be

written as

119878119896 (119903 119905) = int

119905

0

120596119896(1199051015840) 1198891199051015840∓ 1205960 (119905) int

119903

0

119889119903

radic119860119861

119896 = 1 2 3

(28)

Thus from (15) using solutions (27) and (28) into (17) thewave functions for absorption and emission of scalar particlecan be expressed as

120595emm (119903 119905) = expminus 119894ℏ[(int

119905

0

1205960(1199051015840) 1198891199051015840

+Σ119896ℏ119896int

119905

0

120596119896(1199051015840) 1198891199051015840)

minus (1205960 (119905) + Σ119896ℏ

119896120596119896 (119905)) int

119903

0

119889119903

radic119860119861

]

120595abs (119903 119905) = expminus 119894ℏ[(int

119905

0

1205960(1199051015840) 1198891199051015840

+Σ119896ℏ119896int

119905

0

120596119896(1199051015840) 1198891199051015840)

+ (1205960 (119905) + Σ119896ℏ

119896120596119896 (119905)) int

119903

0

119889119903

radic119860119861

]

(29)

respectively Due to tunnelling across the horizon there willbe a change of sign of the metric coefficients in the (119903 119905)-partof the metric and as a result function of 119905 coordinate has animaginary part which will contribute to the probabilities Sowe write

119875abs =1003816100381610038161003816120595abs (119903 119905)

1003816100381610038161003816

2

= exp2 Imℏ

[(int

119905

0

1205960(1199051015840) 1198891199051015840+ Σ119896ℏ119896int

119905

0

120596119896(1199051015840) 1198891199051015840)

+ (1205960 (119905) + Σ119896ℏ

119896120596119896 (119905)) int

119903

0

119889119903

radic119860119861

]

(30)

119875emm =1003816100381610038161003816120595emm (119903 119905)

1003816100381610038161003816

2

= exp2 Imℏ

[(int

119905

0

1205960(1199051015840) 1198891199051015840+ Σ119896ℏ119896int

119905

0

120596119896(1199051015840) 1198891199051015840)

minus (1205960 (119905) + Σ119896ℏ

119896120596119896 (119905)) int

119903

0

119889119903

radic119860119861

]

(31)

To have some simplification we will now use the physical factthat all incoming particles certainly cross the horizon that is119875abs = 1 So from (30)

Im(int

119905

0

1205960(1199051015840) 1198891199051015840+ Σ119896ℏ119896int

119905

0

120596119896(1199051015840) 1198891199051015840)

= minus Im (1205960 (119905) + Σ119896ℏ

119896120596119896 (119905)) int

119903

0

119889119903

radic119860119861

(32)

and hence 119875emm simplifies to

119875emm = expminus4ℏ(1205960 (119905) + Σ119896ℏ

119896120596119896 (119905)) Imint

119903

0

119889119903

radic119860119861

(33)

Advances in High Energy Physics 5

Then from the principle of ldquodetailed balancerdquo [10ndash12] (whichstates that transitions between any two states take place withequal frequency in either direction at equilibrium) we write

119875emm = expminus1205960 (119905)

119879ℎ

119875in = expminus1205960 (119905)

119879ℎ

(34)

So comparing (33) and (34) the temperature of the BH isgiven by

119879ℎ=ℏ

4[1 + Σ

119896ℏ119896120596119896(119905)

1205960(119905)]

minus1

[Imint

119903

0

119889119903

radic119860119861

]

minus1

(35)

where

119879ℎ=ℏ

4[Imint

119903

0

119889119903

radic119860119861

]

minus1

(36)

is the usual Hawking temperature of the BH Thus due toquantum corrections the temperature of the BH is modifiedfrom the Hawking temperature and both temperatures arefunctions of 119905 and 119903 Note that (36) is the standard expressionfor semiclassical Hawking temperature and it is valid fornonspherical metric also However for spherical metric onecan use the Taylor series expansions (7) near the horizonand obtain 119879

119867as given in (11) by performing the contour

integration The ambiguity of factor of two (as mentionedearlier) in the Hawking temperature does not arise here

Further one may note that solutions (27) or (28) are theunique solutions to (18) or (21) except for a premultiplicationfactor This arbitrary multiplicative factor does not appear inthe expression for Hawking temperature only the particleenergy (120596

0) or 120596

119896is rescaled As quantum correction term

contains1205961198961205960 so it does not involve the arbitrarymultiplica-

tive factor and hence it is uniqueTo have some interpretation about the arbitrary functions

120596119896(119905) appearing in the quantum correction terms we make

use of dimensional analysis As 1198780has the dimension ℏ so the

arbitrary function 120596119896(119905) has the dimension ℏminus119896 In standard

choice of units namely 119866 = 119888 = 119870119861= 1 ℏ sim 119872

2

119901and so

120596119896sim 119872minus2119896 where119872 is the mass of the BH

Similar to the Hawking temperature the surface gravityof the BH is modified due to quantum corrections If 120581

119888is

the semiclassical surface gravity corresponding to Hawkingtemperature that is 120581

119888= 2120587119879

119867 then the quantum corrected

surface gravity 120581 = 2120587119879119867is related to the semiclassical value

by the relation

120581 = 120581119888[1 + Σ

119896ℏ119896120596119896(119905)

1205960(119905)]

minus1

(37)

Moreover based on the dimensional analysis if we choosefor simplicity

120596119896 (119905) =

1198861198961205960 (119905)

1198722119896 ldquo119886rdquo is a dimensionless parameter

(38)

then expression (37) is simplified to

120581 = 1205810(1 minus

ℏ119886

1198722)

minus1

(39)

This is related to the one loop back reaction effects in thespacetime [6 33] with the parameter 119886 corresponding to traceanomaly Higher order loop corrections to the surface gravitycan be obtained similarly by suitable choice of the functions120596119896(119905) For static BHs Banerjee and Majhi [19] have studied

these corrections in detail Lastly it is worth mentioning thatidentical result for BH temperature may be obtained if we usethe Painleve coordinate system as in the previous section

4 Entropy Function and Quantum Correction

We will now examine how the semiclassical Bekenstein-Hawking area law namely 119878BH = (1198604ℏ) (119860 is the area of thehorizon) is modified due to quantum corrections describedin the previous section The first law of the BH mechanicswhich is essentially the energy conservation relation relatedthe change of BHmass (119872) to the change of its entropy (119878BH)electric charge (119876) and angular momentum (119869) as

119889119872 = 119879ℎ119889119878BH + Φ119889119876 + Ω119889119869 (40)

Here Ω is the angular velocity and Φ is the electrostaticpotential So for nonrotating uncharged BHs the entropy hasthe simple form

119878BH = int119889119872

119879ℎ

(41)

or using (35) for 119879ℎ we get

119878BH = int[1 + Σ119896ℏ119896120596119896 (119905)

1205960 (119905)

]119889119872

119879119867

(42)

For choice (38) corresponding to one loop back reactioneffects we have from (42) the quantum corrected BH entropyas

119878BH = int[1 +119886ℏ

119872+1198862ℏ2

1198722+ sdot sdot sdot ]

119889119872

119879119867

(43)

The first term is the usual semiclassical Bekenstein-Hawkingentropy and the subsequent terms are the quantum cor-rections of different order For static BHs Banerjee andMajhi [19] have shown the correction terms of which theleading one gives the standard logarithmic correction On theother hand for nonstatic BHs as the proportionality factorsare time-dependent and arbitrary (see (42)) so the leadingorder correction term may not be logarithmic For futurework we will attempt to determine physical interpretation ofthe arbitrary time-dependent proportionality factors so thatquantum corrections may be evaluated

5 Hamilton-Jacobi Method for MassiveParticles Quantum Corrections

The KG equation for a scalar field 120595 describing a scalarparticle of mass119898

0has the form [10]

(◻ +1198982

0

ℏ2)120595 = 0 (44)

6 Advances in High Energy Physics

where the box operator ◻ is evaluated in the background ofa general static BH metric of the form

1198891199042= minus119860 (119903) 119889119905

2+1198891199032

119861 (119903)+ 1199032119889Ω2

2 (45)

The explicit form of the KG equation for the metric (45) is

minus1

119860

1205972120595

1205971199052+ 119861

1205972120595

1205971199032+1

2119860

120597 (119860119861)

120597119903

120597120595

120597119903+2119861

119903

120597120595

120597119903

+1

1199032 sin 120579120597

120597120579(sin 120579

120597120595

120597120579)

+1

1199032sin21205791205972120595

1205971206012=1198982

0

ℏ2120595 (119905 119903 120579 120601)

(46)

Due to spherical symmetry we can decompose 120601 in the form

120595 (119905 119903 120579 120601) = Φ (119905 119903) 119884119898

119897(120579 120601) (47)

where 120601 satisfies [10]

1

119860

1205972120595

1205971199052minus 119861

1205972120595

1205971199032minus1

2119860

120597 (119860119861)

120597119903

120597120595

120597119903minus2119861

119903

120597120595

120597119903

+ 119897 (119897 + 1)

1199032+1198982

0

ℏ2Φ (119905 119903) = 0

(48)

If we substitute the standard ansatz for the semiclassical wavefunction namely

120601 (119905 119903) = exp minus 119894ℏ119878 (119903 119905) (49)

then the action 119878 will satisfy the following differential equa-tion

[1

119860(120597119878

120597119905)

2

minus 119861(120597119878

120597119903)

2

minus 1198642

0(119903)]

minusℏ

119894[1

119860

1205972119878

1205971199052minus 1198612 1205972119878

1205971199032minus

1

2119860

120597 (119860119861)

120597119903+2119861

119903120597119878

120597119903] = 0

(50)

where 11986420= 1198982

0+ (11987121199032) and 1198712 = 119897(119897 + 1)ℏ2 is the angular

momentum To incorporate quantum corrections over thesemiclassical action we expand the actions in powers ofPlanck constant ℏ as

119878 (119903 119905) = 1198780 (119903 119905) + Σ119896ℏ119896119878119896 (119903 119905) (51)

where 1198780is the semiclassical action and 119896 is a positive integer

Now substituting this ansatz for 119878 in the differential equation

(50) and equating different powers of ℏ on both sides weobtain the following set of partial differential equations

ℏ01

119860(120597119878

120597119905)

2

minus 119861(120597119878

120597119903)

2

minus 1198642

0(119903) = 0 (52)

ℏ12

119860

1205971198780

120597119905

1205971198781

120597119905minus 2119861

1205971198780

120597119903

1205971198781

120597119903

minus1

119894[1

119860

12059721198780

1205971199052minus 1198612 12059721198780

1205971199032

minus1

2119860

120597 (119860119861)

120597119903+2119861

1199031205971198780

120597119903] = 0

(53)

ℏ21

119860(1205971198781

120597119905)

2

+2

119860

1205971198780

120597119905

1205971198782

120597119905minus 119861(

1205971198781

120597119903)

2

minus 21198611205971198780

120597119903

1205971198782

120597119903

minus1

119894[1

119860

12059721198781

1205971199052minus 1198612 12059721198781

1205971199032

minus1

2119860

120597 (119860119861)

120597119903+2119861

1199031205971198781

120597119903] = 0

(54)

and so onTo solve the semiclassical action 119878

0 we start with the

standard separable choice [10]

1198780 (119903 119905) = 1205960119905 + 1198630 (119903) (55)

Substituting this choice in (52) we obtain

1198630= plusmnint

119903

0

radic1205962

0minus 1198601198642

0

119860119861119889119903 = plusmn119868

0(say) (56)

where + or minus sign corresponds to absorption or emission ofscalar particle Now substituting this choice for 119878

0in (53) we

have the differential equation for first order corrections 1198781as

1205971198781

120597119905∓ radic119860119861radic1 minus

1198601198642

0

1205962

0

1205971198781

120597119903

∓radic119860119861

119894

[[

[

minus1

119903

radic1 minus1198601198642

0

1205962

+

(120597119860120597119903) (1198642

01205962) minus (2119860119871

21205962

01199033)

4radic1 minus (1198601198642

01205962)

]]

]

= 0

(57)

As before 1198781can be written in separable form as

1198781= 1205961119905 + 1198631 (119903) (58)

Advances in High Energy Physics 7

where

1198631= int

119903

0

119889119903

radic119860119861radic1 minus (1198601198642

01205962

0)

times[[

[

plusmn 1205961minusradic119860119861

119894

times

minus1

119903

radic1 minus1198601198642

0

1205962

+

(120597119860120597119903) (1198642

01205962) minus (2119860119871

21205962

01199033)

4radic1 minus (1198601198642

01205962)

]]

]

= plusmn1198681minus 1198682

(59)

Now due to complicated form if we retain terms up to firstorder quantum corrections that is

119878 = 1198780+ ℏ1198781= (1205960+ ℏ1205961) 119905 + 119863

0+ ℏ1198631 (119903) (60)

then the wave function denoting absorption and emissionsolutions of the KG equation (48) using (49) are of the form

120601abs = exp minus 119894ℏ(1205960+ ℏ1205961119905 + 1198680+ ℏ1198681minus ℏ1198682)

120601emm = exp minus 119894ℏ(1205960+ ℏ1205961119905 minus 1198680+ ℏ1198681minus ℏ1198682)

(61)

It is to be noted that in course of tunnelling across the hori-zon the coordinate nature changes that is more preciselythe signs of themetric coefficients in the (119903 119905)-hyperplane arealteredThus we can interpret this as that the time coordinatehas an imaginary part in crossing the horizon and accordinglythe temporal part has contribution to the probabilities [1933] Thus absorption and emission probabilities are given by

119875abs =1003816100381610038161003816120601in

1003816100381610038161003816

2= exp 2

ℏ(Im 120596

0+ ℏ1205961119905)

+ Im 1198680+ ℏ1198681minus Im ℏ119868

2

(62)

119875emm =1003816100381610038161003816120601out

1003816100381610038161003816

2= exp minus 119894

ℏ(Im 120596

0+ ℏ1205961119905)

minus Im 1198680+ ℏ1198681minus Im ℏ119868

2

(63)

In the classical limit ℏ rarr 0 there is no reflection so allingoing particles should be absorbed and hence [33]

limℏrarr0

119875abs = 1 (64)

So from (62) we must have

Im 1205960119905 = Im 119868

0 Im (120596

1119905 minus 1198682) = Im 119868

1 (65)

and as a result 119875emm simplifies to

119875emm = exp[[

[

minus41205960

ℏ

times Im

int

119903

0

119889119903

radic119860119861

(radic1 minus1198601198642

0

1205962

0

+ℏ (12059611205960)

radic1 minus (1198601198642

01205962

0)

)

]]

]

(66)

Using the principle of ldquodetailed balancerdquo [10 11 20 21]namely

119875emm = expminus 119864119879ℎ

119875in = expminus 119864119879ℎ

(67)

the temperature of the BH is given by

119879ℎ=ℏ119864

41205960

[[

[

Im

int

119903

0

119889119903

radic119860119861

times (radic1 minus1198601198642

0

1205962

0

+ℏ(12059611205960)

radic1 minus (1198601198642

01205962

0)

)

]]

]

minus1

(68)

where the semiclassical Hawking temperature of the BH hasthe expression

119879119867=ℏ119864

41205960

[

[

Imint

119903

0

119889119903

radic119860119861

radic1 minus1198601198642

0

1205962

0

]

]

minus1

(69)

Now to obtain themodified form of the surface gravity of theBH we start with the usual relation between surface gravityand Hawking temperature namely

120581119867= 2120587119879

119867 (70)

where 119879119867is given by (69)

So the quantum corrected surface gravity is given by

120581QC = 2120587119879ℎ (71)

Further for the present nonrotating uncharged static BHsusing the law of BH thermodynamics 119889119872 = 119879

ℎ119889119878 we have

the expression for the entropy of the BH as

119878BH = int41205960

ℏ119864(1 +

ℏ1205961

1205960

)119889119872int

119903

0

119889119903

radic119860119861

(72)

Finally it is easy to see from (68) that near the horizonthe presence of 1198642

0term can be neglected as it is multiplied

8 Advances in High Energy Physics

by themetric coefficient119860Therefore the quantum corrected(up to first order) temperature of the BH (in (68)) reduces to

119879ℎ=ℏ119864

41205960

(1 +ℏ1205961

1205960

)

minus1

[int

119903

0

119889119903

radic119860119861

]

minus1

(73)

and the Hawking temperature (given in (69)) becomes

119879119867=ℏ119864

41205960

[int

119903

0

119889119903

radic119860119861

]

minus1

(74)

So we have

119879ℎ= (1 +

ℏ1205961

1205960

)

minus1

119879119867 (75)

We see that if the energy of the tunnelling particle is chosen as1205960(ie 119864 = 120596

0) and 120596

1= 1205731119872 (for notations see Banerjee

and Majhi [19]) then the Hawking temperature given by(74) is the usual one derived for massless particles andthe quantum corrected temperature 119879

ℎgiven in (75) agrees

with that of Banerjee and Majhi [19] for massless particleTherefore Hawking temperature near the horizon remainsthe same for both massless and nonzero mass tunnellingparticles and it agrees with the claim of Srinivasan andPadmanabhan [10] and Banerjee and Majhi [19] For futurework it will be interesting to calculate the temperature of theBH for tunnelling nonzero mass particle with full quantumcorrections and examine whether the result agrees with thatof Banerjee andMajhi [19] near the horizon Finally it will benice to determine quantum corrected entropy of the BH in aconvenient form

6 Summary of the Work

This work is an attempt to study quantum corrections toHawking radiation of massless particle from a dynamical BHas well as for massive particle from a static BH At firstradial null geodesic tunnelling approach has been used withPainleve-type choice of coordinate system to derive semiclas-sical Hawking temperature Then full quantum mechanicalcalculations have performed writing action in a power seriesof the Planck constant ℏ to evaluate the quantum correc-tions to the Hawking temperature Subsequently quantumcorrected surface gravity has been calculated and it is foundthat one loop back reaction effects in the spacetime can beobtained by suitable choice of the arbitrary functions andparameters Finally an expression for the quantum correctedentropy of the BH has been evaluated It is found that dueto the presence of the arbitrary functions in the expressionfor entropy the leading order quantum correction may notbe logarithmic in nature On the other hand in the caseof Hawking radiation of massive particle from static BH itis found that Hawking temperature near the horizon doesnot depend on the mass term as predicted by Srinivasanand Padmanabhan [10] and Banerjee et al [16ndash18] Forfuture work we will try to find a solution for the partialdifferential equation (18) in a more simple form so that morephysical interpretations can be done from the BHparameters

Also it will be interesting to calculate temperature of theBH for tunnelling nonzero mass particle with full quantumcorrection and examine whether the result agrees with thatof Banerjee andMajhi [19] near the horizon Finally it will benice to determine quantum corrected entropy of the BH in aconvenient form

Conflict of Interests

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

Acknowledgments

The authors are thankful to IUCAA Pune India for theirwarm hospitality and research facilities as the work hasbeen done there during a visit Also Subenoy Chakrabortyacknowledges the UGC-DRS Programme in the Departmentof Mathematics Jadavpur University The author SubhajitSaha is thankful to UGC-BSR Programme of JadavpurUniversity for awarding research fellowship The authors arethankful to Ritabrata Biswas and Nairwita Mazumder fortheir help at the initial part of the work

References

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

[2] J B Hartle and S W Hawking ldquoPath-integral derivation ofblack-hole radiancerdquo Physical Review D vol 13 no 8 pp 2188ndash2203 1976

[3] J D Bekenstein ldquoGeneralized second law of thermodynamicsin black-hole physicsrdquo Physical Review D vol 9 p 2188 1974

[4] T Damour andR Ruffini ldquoBlack-hole evaporation in the Klein-Sauter-Heisenberg-Euler formalismrdquo Physical Review D vol 14p 332 1976

[5] S M Christensen and S A Fulling ldquoTrace anomalies and theHawking effectrdquo Physical ReviewD vol 15 no 8 pp 2088ndash21041977

[6] JW York ldquoBlack hole in thermal equilibriumwith a scalar fieldthe back-reactionrdquo Physical Review D vol 31 p 755 1985

[7] Z Zhao and J Y Zhu ldquoNernst theorem and Planck absoluteentropy of black holesrdquo Acta Physica Sinica vol 48 no 8 pp1558ndash1564 1999

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

[9] M K Parikh ldquoEnergy conservation and Hawking radiationrdquohttparxivorgabshep-th0402166

[10] K Srinivasan and T Padmanabhan ldquoParticle production andcomplex path analysisrdquo Physical Review D vol 60 no 2 ArticleID 024007 p 20 1999

[11] S Shankaranarayanan K Srinivasan and T PadmanabhanldquoMethod of complex paths and general covariance of HawkingradiationrdquoModern Physics Letters A Particles and Fields Grav-itation Cosmology Nuclear Physics vol 16 no 9 pp 571ndash5782001

[12] R Banerjee B R Majhi and S Samanta ldquoNoncommutativeblack hole thermodynamicsrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 77 no 12 Article ID124035 p 8 2008

Advances in High Energy Physics 9

[13] Q-Q Jiang S-Q Wu and X Cai ldquoHawking radiation as tun-neling from the Kerr and Kerr-Newman black holesrdquo PhysicalReview D vol 73 no 6 Article ID 064003 p 10 2006

[14] Y Hu J Zhang and Z Zhao ldquoMassive particlesrsquo Hawkingradiation via tunneling from the G H dilaton black holerdquoModern Physics Letters A Particles and Fields GravitationCosmology Nuclear Physics vol 21 no 28 pp 2143ndash2149 2006

[15] Z Xu and B Chen ldquoHawking radiation from general Kerr-(anti)de Sitter black holesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 75 no 2 Article ID 024041 p6 2007

[16] R Kerner and R B Mann ldquoFermions tunnelling from blackholesrdquo Classical and Quantum Gravity vol 25 no 9 Article ID095014 p 17 2008

[17] R D Kerner and R B Mann ldquoCharged fermions tunnellingfrom Kerr-Newman black holesrdquo Physics Letters B vol 665 no4 pp 277ndash283 2008

[18] D-Y Chen Q-Q Jiang and X-T Zu ldquoHawking radiation ofDirac particles via tunnelling from rotating black holes in deSitter spacesrdquo Physics Letters B vol 665 no 2-3 pp 106ndash1102008

[19] R Banerjee and B R Majhi ldquoQuantum tunneling beyondsemiclassical approximationrdquo Journal of High Energy Physicsno 6 Article ID 095 p 20 2008

[20] C Corda ldquoEffective temperature for black holesrdquo Journal ofHigh Energy Physics vol 1108 p 101 2011

[21] C Corda S H Hendi R Katebi and N O Schmidt ldquoEffectivestate Hawking radiation and quasi-normal modes for Kerrblack holesrdquo Journal of High Energy Physics no 6 p 008 2013

[22] H M Siahaan and T Triyanta ldquoSemiclassical methods forhawking radiation from a vaidya black holerdquo InternationalJournal of Modern Physics A vol 25 no 1 pp 145ndash153 2010

[23] B D Chowdhury ldquoProblems with tunneling of thin shells fromblack holesrdquo Pramana vol 70 no 1 pp 593ndash612 2008

[24] E T Akhmedov V Akhmedova and D Singleton ldquoHawkingtemperature in the tunneling picturerdquo Physics Letters B vol 642no 1-2 pp 124ndash128 2006

[25] E T Akhmedov T Pilling and D Singleton ldquoSubtleties in thequasi-classical calculation of Hawking radiationrdquo InternationalJournal of Modern Physics D Gravitation Astrophysics Cosmol-ogy vol 17 no 13-14 pp 2453ndash2458 2008

[26] R Banerjee and B R Majhi ldquoQuantum tunnelingand backreactionrdquo Physics Letters B vol 662 no 1 pp 62ndash65 2008

[27] C Corda ldquoEffective temperature Hawking radiation and quasi-normal modesrdquo International Journal of Modern Physics D vol21 no 11 Article ID 1242023 2012

[28] H-L Li and S-Z Yang ldquoHawking radiation from the chargedBTZ black hole with back-reactionrdquo Europhysics Letters vol 79no 2 Article ID 20001 p 3 2007

[29] C Corda S H Hendi R Katebi and N O Schmidt ldquoHawk-ing radiation-quasi-normal modes correspondence and effec-tive states for nonextremal Reissner-Nordstrom black holesrdquoAdvances in High Energy Physics vol 2014 Article ID 5278749 pages 2014

[30] Q-Q Jiang and S-Q Wu ldquoHawking radiation of chargedparticles as tunneling from Reissner-Nordstrom-de Sitter blackholes with a global monopolerdquo Physics Letters B vol 635 no2-3 pp 151ndash155 2006

[31] Y Hu J Zhang and Z Zhao ldquoMassive uncharged and chargedparticlesrsquo tunneling from the Horowitz-Strominger dilatonblack holerdquo International Journal of Modern Physics D Gravi-tation Astrophysics Cosmology vol 16 no 5 pp 847ndash855 2007

[32] C O Lousto and N G Sanchez ldquoBack reaction effects in blackhole spacetimes rdquo Physics Letters B vol 212 no 4 pp 411ndash4141988

[33] P Mitra ldquoHawking temperature from tunnelling formalismrdquoPhysics Letters B vol 648 no 2-3 pp 240ndash242 2007

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

2 Advances in High Energy Physics

the calculation ofHawking temperature beyond the semiclas-sical limit Mostly both groups have considered tunnellingof massless particle and evaluated the modified Hawkingtemperature with quantum corrections

In the present work at first we consider a generalnonstatic metric for dynamical BH HJ method is extendedbeyond semiclassical approximation to consider all the termsin the expansion of the one particle action It is foundthat the higher order terms (quantum corrections) satisfyidentical differential equations as the semiclassical actionand the complicated terms are eliminated considering BHhorizon as one way barrier We derive the modified Hawkingtemperature using both the above approaches which arefound to be identical at the semiclassical level Also modifiedform of the BH entropy with quantum correction has beenevaluated

Subsequently in the next section we consider tun-nelling of particles having nonzeromass beyond semiclassicalapproximation Due to nonzero mass the imaginary partof the action cannot be evaluated using first approach onlyHJ method will be applicable Further the complicated formof the equations involved restricted us to only first orderquantum correction

2 Method of Radial Null GeodesicA Survey of Earlier Works

This section deals with a brief survey of the method of radialnull geodesicsmethod [8] considering the picture ofHawkingradiation as quantum tunnelling In a word the methodcorrelates the imaginary part of the action for the classicallyforbidden process of s-wave emission across the horizonwith the Boltzmann factor for the black body radiation atthe Hawking temperature We start with a general class ofnonstatic spherically symmetric BH metric of the form

1198891199042= minus119860 (119903 119905) 119889119905

2+

1198891199032

119861 (119903 119905)+ 1199032119889Ω2

2 (2)

where the horizon 119903ℎis located at 119860(119903

ℎ 119905) = 0 = 119861(119903

ℎ 119905)

and the metric has a coordinate singularity at the horizonTo remove the coordinate singularity we make the followingPainleve-type transformation of coordinates

119889119905 997888rarr 119889119905 minus radic1 minus 119861

119860119861119889119903 (3)

and as a result metric (2) transforms to

1198891199042= minus119860119889119905

2+ 2radic119860(

1

119861minus 1)119889119905119889119903 + 119889119903

2+ 1199032119889Ω2

2 (4)

This metric (ie the choice of coordinates) has some distinctfeatures over the former one as follows

(i) The metric is singularity free across the horizon(ii) At any fixed time we have a flat spatial geometry(iii) Both the metric will have the same boundary geome-

try at any fixed radius

The radial null geodesic (characterized by 1198891199042 = 0 =

119889Ω2

2) has the differential equation (using (3))

119889119903

119889119905= radic

119860

119861[plusmn1 minus radic1 minus 119861 (119903 119905)] (5)

where outgoing or ingoing geodesic is identified by the + orminus sign within the square bracket in (4) In the present casewe deal with the absorption of particles through the horizon(ie + sign only) and according to Parikh and Wilczek [8]the imaginary part of the action is obtained as

Im 119878 = Imint

119903out

119903in

119901119903119889119903 = Imint

119903out

119903in

int

119901119903

0

1198891199011015840

119903119889119903

= Imint

119903out

119903in

int

119867

0

1198891198671015840

119889119903119889119905 119889119903

(6)

Note that in the last step of the above derivation we haveused the Hamiltonrsquos equation 119903 = (119889119867119889119901

119903)|119903 where (119903119901

119903)

are canonical pair Further it is to be mentioned that inquantum mechanics the action of a tunnelled particle in apotential barrier having energy larger than the energy of theparticle will be imaginary as 119901

119903= radic2119898(119864 minus 119881) For the

present nonstatic BH the mass of the BH is not constantand hence the 1198891198671015840 integration extends over all the values ofenergy of outgoing particle fromzero to119864(119905) [22] (say) Asweare dealing with tunnelling across the BH horizon so usingTaylor series expansion about the horizon 119903

ℎwe write

119860(119903 119905)|119905 =120597119860(119903 119905)

120597119903

10038161003816100381610038161003816100381610038161003816119905

(119903 minus 119903ℎ) + 119874(119903 minus 119903

ℎ)210038161003816100381610038161003816119905

119861(119903 119905)|119905 =120597119861(119903 119905)

120597119903

10038161003816100381610038161003816100381610038161003816119905

(119903 minus 119903ℎ) + 119874(119903 minus 119903

ℎ)210038161003816100381610038161003816119905

(7)

So in the neighbourhood of the horizon the geodesicequation (4) can be approximated as

119889119903

119889119905asymp1

2radic1198601015840 (119903

ℎ 119905) 1198611015840 (119903

ℎ 119905) (119903 minus 119903

ℎ) (8)

Substituting this value of 119889119903119889119905 in the last step of (5) we have

Im 119878 =2120587119864 (119905)

radic1198601015840 (119903ℎ 119905) 1198611015840 (119903

ℎ 119905)

(9)

where the choice of contour for 119903-integration is on the upperhalf complex plane to avoid the coordinate singularity at 119903

ℎ

Thus the tunnelling probability is given by

Γ sim exp minus2ℏIm 119878 = expminus 4120587119864 (119905)

ℏradic11986010158401198611015840 (10)

which in turn equateswith the Boltzmann factor exp119864(119905)119879the expression for the Hawking temperature is

119879119867=

ℏradic1198601015840 (119903ℎ 119905) 1198611015840 (119903

ℎ 119905)

4120587

(11)

Advances in High Energy Physics 3

From the above expression it is to be noted that 119879119867is time

dependentRecently a drawback of the above approach

has been noted [23ndash25] It has been shown thatΓ sim expminus(2ℏ) Im 119878 = expminus(2ℏ) Imint

119903out

119903in119901119903119889119903 is not

canonically invariant and hence is not a proper observable itshould be modified as expminus Im∮119901

119903119889119903ℏ The closed path

goes across the horizon and back For tunnelling across theordinary barrier it is immaterial whether the particle goesfrom the left to the right or the reverse path So in that case

∮119901119903119889119903 = 2int

119903out

119903in

119901119903119889119903 (12)

and there is no problem of canonical invariance But difficultyarises for BH horizon which behaves as a barrier for particlesgoing from inside of the BH to outside but it does not actas a barrier for particles going from outside to the insideSo relation (12) is no longer valid Also using tunnelling theprobability is Γ sim expminus Im∮119901

119903119889119903ℏ so there will be a

problem of factor two in Hawking temperature [24 26 27]Further the above analysis of tunnelling approach

remains incomplete unless effects of self-gravitation and backreaction are taken into account But unfortunately no generalapproaches to account for the above effects are there in theliterature only few results are available for some known BHsolutions [26ndash32]

Finally it is worth mentioning that so far the abovetunnelling approach is purely semiclassical in nature andquantum corrections are not included Also this method isapplicable for Painleve-type coordinates only one cannot usethe original metric coordinates to avoid horizon singularityLastly the tunnelling approach is not applicable for massiveparticles [19]

3 Hamilton-Jacobi MethodQuantum Corrections

We will now follow the alternative approach as mentioned inthe introduction that is the HJ method to evaluate the imag-inary part of the action and hence the Hawking temperatureWe will analyze the beyond semiclassical approximation byincorporating possible quantum corrections As this methodis not affected by the coordinate singularity at the horizon sowe will use the general BH metric (2) for convenience

In the background of the gravitational field describedby the metric (2) massless scalar particles obey the Klein-Gordon equation

ℏ2

radicminus119892120597 [119892120583]radicminus119892120597]] 120595 = 0 (13)

For spherically symmetric BH as we are only consideringradial trajectories so we will consider (119905 119903)-sector in thespacetime given by (2) that is we concentrate on two-dimensional BH problems Using (2) the above Klein-Gordon equation becomes

1205972120595

1205971199052minus

1

2119860119861

120597 (119860119861)

120597119905

120597120595

120597119905minus1

2

120597 (119860119861)

120597119903

120597120595

120597119903minus 119860119861

1205972120595

1205971199032= 0 (14)

Using the standard ansatz for the semiclassical wave functionnamely

120595 (119903 119905) = exp minus 119894ℏ119878 (119903 119905) (15)

the differential equation for the action 119878 is

(120597119878

120597119905)

2

minus 119860119861(120597119878

120597119903)

2

+ 119894ℏ [1205972119878

1205971199052minus

1

2119860119861

120597 (119860119861)

120597119905

120597119878

120597119905minus1

2

120597 (119860119861)

120597119903

120597119878

120597119903minus 119860119861

1205972119878

1205971199032]

(16)

To solve this partial differential equation we expand theaction 119878 in powers of Planckrsquos constant ℏ as

119878 (119903 119905) = 1198780 (119903 119905) + Σℏ119896119878119896 (119903 119905) (17)

with 119896 being a positive integer Note that in the aboveexpansion terms of the order of Planckrsquos constant and itshigher powers are considered as quantum corrections overthe semiclassical action 119878

0 Now substituting ansatz (17) for 119878

into (16) and equating different powers of ℏ on both sides weobtain the following set of partial differential equations

ℏ0 (120597119878

120597119905)

2

minus 119860119861(120597119878

120597119903)

2

= 0 (18)

ℏ11205971198780

120597119905

1205971198781

120597119905minus 119860119861

1205971198780

120597119903

1205971198781

120597119903

+119894

2[12059721198780

1205971199052minus

1

2119860119861

120597 (119860119861)

120597119905

1205971198780

120597119905

minus1

2

120597 (119860119861)

120597119903

1205971198780

120597119903minus 119860119861

12059721198780

1205971199032] = 0

(19)

ℏ2 (1205971198781

120597119905)

2

+ 21205971198780

120597119905

1205971198782

120597119905minus 119860119861(

1205971198781

120597119903)

2

minus 21198601198611205971198780

120597119903

1205971198782

120597119903

+ 119894 [12059721198781

1205971199052minus

1

2119860119861

120597 (119860119861)

120597119905

1205971198781

120597119905

minus1

2

120597 (119860119861)

120597119903

1205971198781

120597119903minus 119860119861

12059721198781

1205971199032] = 0

(20)

and so onApparently different order partial differential equations

are very complicated but fortunately there will be lot ofsimplifications if in the partial differential equation corre-sponding to ℏ119896 all previous partial differential equationsare used and finally we obtain identical partial differentialequation namely

ℏ119896120597119878119896

120597119905= plusmnradic119860 (119903 119905) 119861 (119903 119905)

120597119878119896

120597119903 (21)

for 119896 = 0 1 2

4 Advances in High Energy Physics

Thus quantum corrections satisfy the same differentialequation as the semiclassical action 119878

0 Hence the solutions

will be very similar To solve 1198780 it is to be noted that due to

nonstatic BHs the metric coefficients are functions of 119903 and119905 and hence standard HJ method cannot be applied somegeneralization is needed We start with a general metric [22]

1198780 (119903 119905) = int

119905

0

1205960(1199051015840) 119889119905 + 119863

0 (119903 119905) (22)

Here 1205960(119905) behaves as the energy of the emitted particle

and the justification of the choice of the integral is that theoutgoing particle should have time-dependent continuumenergy

Now substituting the above ansatz for 1198780(119903 119905) into (18) and

using the radial null geodesic in the usual metric from (2)namely

119889119903

119889119905= plusmnradic119860119861 (23)

we have

1205971198630

120597119903+1205971198630

120597119905

119889119905

119889119903= ∓1205960 (119905)

119889119905

119889119903 (24)

that is

1198891198630

119889119903= ∓

1205960 (119905)

radic119860119861

(25)

which gives

1198630= ∓1205960 (119905) int

119903

0

119889119903

radic119860119861

(26)

Hence the complete semiclassical action takes the form

1198780 (119903 119905) = int

119905

0

1205960(1199051015840) 1198891199051015840∓ 1205960 (119905) int

119903

0

119889119903

radic119860119861

(27)

Here the minus (or +) sign corresponds to absorption (or emis-sion) particle As solution (27) contains an arbitrary time-dependent function 120596

0(119905) so a general solution for 119878

119896can be

written as

119878119896 (119903 119905) = int

119905

0

120596119896(1199051015840) 1198891199051015840∓ 1205960 (119905) int

119903

0

119889119903

radic119860119861

119896 = 1 2 3

(28)

Thus from (15) using solutions (27) and (28) into (17) thewave functions for absorption and emission of scalar particlecan be expressed as

120595emm (119903 119905) = expminus 119894ℏ[(int

119905

0

1205960(1199051015840) 1198891199051015840

+Σ119896ℏ119896int

119905

0

120596119896(1199051015840) 1198891199051015840)

minus (1205960 (119905) + Σ119896ℏ

119896120596119896 (119905)) int

119903

0

119889119903

radic119860119861

]

120595abs (119903 119905) = expminus 119894ℏ[(int

119905

0

1205960(1199051015840) 1198891199051015840

+Σ119896ℏ119896int

119905

0

120596119896(1199051015840) 1198891199051015840)

+ (1205960 (119905) + Σ119896ℏ

119896120596119896 (119905)) int

119903

0

119889119903

radic119860119861

]

(29)

respectively Due to tunnelling across the horizon there willbe a change of sign of the metric coefficients in the (119903 119905)-partof the metric and as a result function of 119905 coordinate has animaginary part which will contribute to the probabilities Sowe write

119875abs =1003816100381610038161003816120595abs (119903 119905)

1003816100381610038161003816

2

= exp2 Imℏ

[(int

119905

0

1205960(1199051015840) 1198891199051015840+ Σ119896ℏ119896int

119905

0

120596119896(1199051015840) 1198891199051015840)

+ (1205960 (119905) + Σ119896ℏ

119896120596119896 (119905)) int

119903

0

119889119903

radic119860119861

]

(30)

119875emm =1003816100381610038161003816120595emm (119903 119905)

1003816100381610038161003816

2

= exp2 Imℏ

[(int

119905

0

1205960(1199051015840) 1198891199051015840+ Σ119896ℏ119896int

119905

0

120596119896(1199051015840) 1198891199051015840)

minus (1205960 (119905) + Σ119896ℏ

119896120596119896 (119905)) int

119903

0

119889119903

radic119860119861

]

(31)

To have some simplification we will now use the physical factthat all incoming particles certainly cross the horizon that is119875abs = 1 So from (30)

Im(int

119905

0

1205960(1199051015840) 1198891199051015840+ Σ119896ℏ119896int

119905

0

120596119896(1199051015840) 1198891199051015840)

= minus Im (1205960 (119905) + Σ119896ℏ

119896120596119896 (119905)) int

119903

0

119889119903

radic119860119861

(32)

and hence 119875emm simplifies to

119875emm = expminus4ℏ(1205960 (119905) + Σ119896ℏ

119896120596119896 (119905)) Imint

119903

0

119889119903

radic119860119861

(33)

Advances in High Energy Physics 5

Then from the principle of ldquodetailed balancerdquo [10ndash12] (whichstates that transitions between any two states take place withequal frequency in either direction at equilibrium) we write

119875emm = expminus1205960 (119905)

119879ℎ

119875in = expminus1205960 (119905)

119879ℎ

(34)

So comparing (33) and (34) the temperature of the BH isgiven by

119879ℎ=ℏ

4[1 + Σ

119896ℏ119896120596119896(119905)

1205960(119905)]

minus1

[Imint

119903

0

119889119903

radic119860119861

]

minus1

(35)

where

119879ℎ=ℏ

4[Imint

119903

0

119889119903

radic119860119861

]

minus1

(36)

is the usual Hawking temperature of the BH Thus due toquantum corrections the temperature of the BH is modifiedfrom the Hawking temperature and both temperatures arefunctions of 119905 and 119903 Note that (36) is the standard expressionfor semiclassical Hawking temperature and it is valid fornonspherical metric also However for spherical metric onecan use the Taylor series expansions (7) near the horizonand obtain 119879

119867as given in (11) by performing the contour

integration The ambiguity of factor of two (as mentionedearlier) in the Hawking temperature does not arise here

Further one may note that solutions (27) or (28) are theunique solutions to (18) or (21) except for a premultiplicationfactor This arbitrary multiplicative factor does not appear inthe expression for Hawking temperature only the particleenergy (120596

0) or 120596

119896is rescaled As quantum correction term

contains1205961198961205960 so it does not involve the arbitrarymultiplica-

tive factor and hence it is uniqueTo have some interpretation about the arbitrary functions

120596119896(119905) appearing in the quantum correction terms we make

use of dimensional analysis As 1198780has the dimension ℏ so the

arbitrary function 120596119896(119905) has the dimension ℏminus119896 In standard

choice of units namely 119866 = 119888 = 119870119861= 1 ℏ sim 119872

2

119901and so

120596119896sim 119872minus2119896 where119872 is the mass of the BH

Similar to the Hawking temperature the surface gravityof the BH is modified due to quantum corrections If 120581

119888is

the semiclassical surface gravity corresponding to Hawkingtemperature that is 120581

119888= 2120587119879

119867 then the quantum corrected

surface gravity 120581 = 2120587119879119867is related to the semiclassical value

by the relation

120581 = 120581119888[1 + Σ

119896ℏ119896120596119896(119905)

1205960(119905)]

minus1

(37)

Moreover based on the dimensional analysis if we choosefor simplicity

120596119896 (119905) =

1198861198961205960 (119905)

1198722119896 ldquo119886rdquo is a dimensionless parameter

(38)

then expression (37) is simplified to

120581 = 1205810(1 minus

ℏ119886

1198722)

minus1

(39)

This is related to the one loop back reaction effects in thespacetime [6 33] with the parameter 119886 corresponding to traceanomaly Higher order loop corrections to the surface gravitycan be obtained similarly by suitable choice of the functions120596119896(119905) For static BHs Banerjee and Majhi [19] have studied

these corrections in detail Lastly it is worth mentioning thatidentical result for BH temperature may be obtained if we usethe Painleve coordinate system as in the previous section

4 Entropy Function and Quantum Correction

We will now examine how the semiclassical Bekenstein-Hawking area law namely 119878BH = (1198604ℏ) (119860 is the area of thehorizon) is modified due to quantum corrections describedin the previous section The first law of the BH mechanicswhich is essentially the energy conservation relation relatedthe change of BHmass (119872) to the change of its entropy (119878BH)electric charge (119876) and angular momentum (119869) as

119889119872 = 119879ℎ119889119878BH + Φ119889119876 + Ω119889119869 (40)

Here Ω is the angular velocity and Φ is the electrostaticpotential So for nonrotating uncharged BHs the entropy hasthe simple form

119878BH = int119889119872

119879ℎ

(41)

or using (35) for 119879ℎ we get

119878BH = int[1 + Σ119896ℏ119896120596119896 (119905)

1205960 (119905)

]119889119872

119879119867

(42)

For choice (38) corresponding to one loop back reactioneffects we have from (42) the quantum corrected BH entropyas

119878BH = int[1 +119886ℏ

119872+1198862ℏ2

1198722+ sdot sdot sdot ]

119889119872

119879119867

(43)

The first term is the usual semiclassical Bekenstein-Hawkingentropy and the subsequent terms are the quantum cor-rections of different order For static BHs Banerjee andMajhi [19] have shown the correction terms of which theleading one gives the standard logarithmic correction On theother hand for nonstatic BHs as the proportionality factorsare time-dependent and arbitrary (see (42)) so the leadingorder correction term may not be logarithmic For futurework we will attempt to determine physical interpretation ofthe arbitrary time-dependent proportionality factors so thatquantum corrections may be evaluated

5 Hamilton-Jacobi Method for MassiveParticles Quantum Corrections

The KG equation for a scalar field 120595 describing a scalarparticle of mass119898

0has the form [10]

(◻ +1198982

0

ℏ2)120595 = 0 (44)

6 Advances in High Energy Physics

where the box operator ◻ is evaluated in the background ofa general static BH metric of the form

1198891199042= minus119860 (119903) 119889119905

2+1198891199032

119861 (119903)+ 1199032119889Ω2

2 (45)

The explicit form of the KG equation for the metric (45) is

minus1

119860

1205972120595

1205971199052+ 119861

1205972120595

1205971199032+1

2119860

120597 (119860119861)

120597119903

120597120595

120597119903+2119861

119903

120597120595

120597119903

+1

1199032 sin 120579120597

120597120579(sin 120579

120597120595

120597120579)

+1

1199032sin21205791205972120595

1205971206012=1198982

0

ℏ2120595 (119905 119903 120579 120601)

(46)

Due to spherical symmetry we can decompose 120601 in the form

120595 (119905 119903 120579 120601) = Φ (119905 119903) 119884119898

119897(120579 120601) (47)

where 120601 satisfies [10]

1

119860

1205972120595

1205971199052minus 119861

1205972120595

1205971199032minus1

2119860

120597 (119860119861)

120597119903

120597120595

120597119903minus2119861

119903

120597120595

120597119903

+ 119897 (119897 + 1)

1199032+1198982

0

ℏ2Φ (119905 119903) = 0

(48)

If we substitute the standard ansatz for the semiclassical wavefunction namely

120601 (119905 119903) = exp minus 119894ℏ119878 (119903 119905) (49)

then the action 119878 will satisfy the following differential equa-tion

[1

119860(120597119878

120597119905)

2

minus 119861(120597119878

120597119903)

2

minus 1198642

0(119903)]

minusℏ

119894[1

119860

1205972119878

1205971199052minus 1198612 1205972119878

1205971199032minus

1

2119860

120597 (119860119861)

120597119903+2119861

119903120597119878

120597119903] = 0

(50)

where 11986420= 1198982

0+ (11987121199032) and 1198712 = 119897(119897 + 1)ℏ2 is the angular

momentum To incorporate quantum corrections over thesemiclassical action we expand the actions in powers ofPlanck constant ℏ as

119878 (119903 119905) = 1198780 (119903 119905) + Σ119896ℏ119896119878119896 (119903 119905) (51)

where 1198780is the semiclassical action and 119896 is a positive integer

Now substituting this ansatz for 119878 in the differential equation

(50) and equating different powers of ℏ on both sides weobtain the following set of partial differential equations

ℏ01

119860(120597119878

120597119905)

2

minus 119861(120597119878

120597119903)

2

minus 1198642

0(119903) = 0 (52)

ℏ12

119860

1205971198780

120597119905

1205971198781

120597119905minus 2119861

1205971198780

120597119903

1205971198781

120597119903

minus1

119894[1

119860

12059721198780

1205971199052minus 1198612 12059721198780

1205971199032

minus1

2119860

120597 (119860119861)

120597119903+2119861

1199031205971198780

120597119903] = 0

(53)

ℏ21

119860(1205971198781

120597119905)

2

+2

119860

1205971198780

120597119905

1205971198782

120597119905minus 119861(

1205971198781

120597119903)

2

minus 21198611205971198780

120597119903

1205971198782

120597119903

minus1

119894[1

119860

12059721198781

1205971199052minus 1198612 12059721198781

1205971199032

minus1

2119860

120597 (119860119861)

120597119903+2119861

1199031205971198781

120597119903] = 0

(54)

and so onTo solve the semiclassical action 119878

0 we start with the

standard separable choice [10]

1198780 (119903 119905) = 1205960119905 + 1198630 (119903) (55)

Substituting this choice in (52) we obtain

1198630= plusmnint

119903

0

radic1205962

0minus 1198601198642

0

119860119861119889119903 = plusmn119868

0(say) (56)

where + or minus sign corresponds to absorption or emission ofscalar particle Now substituting this choice for 119878

0in (53) we

have the differential equation for first order corrections 1198781as

1205971198781

120597119905∓ radic119860119861radic1 minus

1198601198642

0

1205962

0

1205971198781

120597119903

∓radic119860119861

119894

[[

[

minus1

119903

radic1 minus1198601198642

0

1205962

+

(120597119860120597119903) (1198642

01205962) minus (2119860119871

21205962

01199033)

4radic1 minus (1198601198642

01205962)

]]

]

= 0

(57)

As before 1198781can be written in separable form as

1198781= 1205961119905 + 1198631 (119903) (58)

Advances in High Energy Physics 7

where

1198631= int

119903

0

119889119903

radic119860119861radic1 minus (1198601198642

01205962

0)

times[[

[

plusmn 1205961minusradic119860119861

119894

times

minus1

119903

radic1 minus1198601198642

0

1205962

+

(120597119860120597119903) (1198642

01205962) minus (2119860119871

21205962

01199033)

4radic1 minus (1198601198642

01205962)

]]

]

= plusmn1198681minus 1198682

(59)

Now due to complicated form if we retain terms up to firstorder quantum corrections that is

119878 = 1198780+ ℏ1198781= (1205960+ ℏ1205961) 119905 + 119863

0+ ℏ1198631 (119903) (60)

then the wave function denoting absorption and emissionsolutions of the KG equation (48) using (49) are of the form

120601abs = exp minus 119894ℏ(1205960+ ℏ1205961119905 + 1198680+ ℏ1198681minus ℏ1198682)

120601emm = exp minus 119894ℏ(1205960+ ℏ1205961119905 minus 1198680+ ℏ1198681minus ℏ1198682)

(61)

It is to be noted that in course of tunnelling across the hori-zon the coordinate nature changes that is more preciselythe signs of themetric coefficients in the (119903 119905)-hyperplane arealteredThus we can interpret this as that the time coordinatehas an imaginary part in crossing the horizon and accordinglythe temporal part has contribution to the probabilities [1933] Thus absorption and emission probabilities are given by

119875abs =1003816100381610038161003816120601in

1003816100381610038161003816

2= exp 2

ℏ(Im 120596

0+ ℏ1205961119905)

+ Im 1198680+ ℏ1198681minus Im ℏ119868

2

(62)

119875emm =1003816100381610038161003816120601out

1003816100381610038161003816

2= exp minus 119894

ℏ(Im 120596

0+ ℏ1205961119905)

minus Im 1198680+ ℏ1198681minus Im ℏ119868

2

(63)

In the classical limit ℏ rarr 0 there is no reflection so allingoing particles should be absorbed and hence [33]

limℏrarr0

119875abs = 1 (64)

So from (62) we must have

Im 1205960119905 = Im 119868

0 Im (120596

1119905 minus 1198682) = Im 119868

1 (65)

and as a result 119875emm simplifies to

119875emm = exp[[

[

minus41205960

ℏ

times Im

int

119903

0

119889119903

radic119860119861

(radic1 minus1198601198642

0

1205962

0

+ℏ (12059611205960)

radic1 minus (1198601198642

01205962

0)

)

]]

]

(66)

Using the principle of ldquodetailed balancerdquo [10 11 20 21]namely

119875emm = expminus 119864119879ℎ

119875in = expminus 119864119879ℎ

(67)

the temperature of the BH is given by

119879ℎ=ℏ119864

41205960

[[

[

Im

int

119903

0

119889119903

radic119860119861

times (radic1 minus1198601198642

0

1205962

0

+ℏ(12059611205960)

radic1 minus (1198601198642

01205962

0)

)

]]

]

minus1

(68)

where the semiclassical Hawking temperature of the BH hasthe expression

119879119867=ℏ119864

41205960

[

[

Imint

119903

0

119889119903

radic119860119861

radic1 minus1198601198642

0

1205962

0

]

]

minus1

(69)

Now to obtain themodified form of the surface gravity of theBH we start with the usual relation between surface gravityand Hawking temperature namely

120581119867= 2120587119879

119867 (70)

where 119879119867is given by (69)

So the quantum corrected surface gravity is given by

120581QC = 2120587119879ℎ (71)

Further for the present nonrotating uncharged static BHsusing the law of BH thermodynamics 119889119872 = 119879

ℎ119889119878 we have

the expression for the entropy of the BH as

119878BH = int41205960

ℏ119864(1 +

ℏ1205961

1205960

)119889119872int

119903

0

119889119903

radic119860119861

(72)

Finally it is easy to see from (68) that near the horizonthe presence of 1198642

0term can be neglected as it is multiplied

8 Advances in High Energy Physics

by themetric coefficient119860Therefore the quantum corrected(up to first order) temperature of the BH (in (68)) reduces to

119879ℎ=ℏ119864

41205960

(1 +ℏ1205961

1205960

)

minus1

[int

119903

0

119889119903

radic119860119861

]

minus1

(73)

and the Hawking temperature (given in (69)) becomes

119879119867=ℏ119864

41205960

[int

119903

0

119889119903

radic119860119861

]

minus1

(74)

So we have

119879ℎ= (1 +

ℏ1205961

1205960

)

minus1

119879119867 (75)

We see that if the energy of the tunnelling particle is chosen as1205960(ie 119864 = 120596

0) and 120596

1= 1205731119872 (for notations see Banerjee

and Majhi [19]) then the Hawking temperature given by(74) is the usual one derived for massless particles andthe quantum corrected temperature 119879

ℎgiven in (75) agrees

with that of Banerjee and Majhi [19] for massless particleTherefore Hawking temperature near the horizon remainsthe same for both massless and nonzero mass tunnellingparticles and it agrees with the claim of Srinivasan andPadmanabhan [10] and Banerjee and Majhi [19] For futurework it will be interesting to calculate the temperature of theBH for tunnelling nonzero mass particle with full quantumcorrections and examine whether the result agrees with thatof Banerjee andMajhi [19] near the horizon Finally it will benice to determine quantum corrected entropy of the BH in aconvenient form

6 Summary of the Work

This work is an attempt to study quantum corrections toHawking radiation of massless particle from a dynamical BHas well as for massive particle from a static BH At firstradial null geodesic tunnelling approach has been used withPainleve-type choice of coordinate system to derive semiclas-sical Hawking temperature Then full quantum mechanicalcalculations have performed writing action in a power seriesof the Planck constant ℏ to evaluate the quantum correc-tions to the Hawking temperature Subsequently quantumcorrected surface gravity has been calculated and it is foundthat one loop back reaction effects in the spacetime can beobtained by suitable choice of the arbitrary functions andparameters Finally an expression for the quantum correctedentropy of the BH has been evaluated It is found that dueto the presence of the arbitrary functions in the expressionfor entropy the leading order quantum correction may notbe logarithmic in nature On the other hand in the caseof Hawking radiation of massive particle from static BH itis found that Hawking temperature near the horizon doesnot depend on the mass term as predicted by Srinivasanand Padmanabhan [10] and Banerjee et al [16ndash18] Forfuture work we will try to find a solution for the partialdifferential equation (18) in a more simple form so that morephysical interpretations can be done from the BHparameters

Also it will be interesting to calculate temperature of theBH for tunnelling nonzero mass particle with full quantumcorrection and examine whether the result agrees with thatof Banerjee andMajhi [19] near the horizon Finally it will benice to determine quantum corrected entropy of the BH in aconvenient form

Conflict of Interests

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

Acknowledgments

The authors are thankful to IUCAA Pune India for theirwarm hospitality and research facilities as the work hasbeen done there during a visit Also Subenoy Chakrabortyacknowledges the UGC-DRS Programme in the Departmentof Mathematics Jadavpur University The author SubhajitSaha is thankful to UGC-BSR Programme of JadavpurUniversity for awarding research fellowship The authors arethankful to Ritabrata Biswas and Nairwita Mazumder fortheir help at the initial part of the work

References

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

[2] J B Hartle and S W Hawking ldquoPath-integral derivation ofblack-hole radiancerdquo Physical Review D vol 13 no 8 pp 2188ndash2203 1976

[3] J D Bekenstein ldquoGeneralized second law of thermodynamicsin black-hole physicsrdquo Physical Review D vol 9 p 2188 1974

[4] T Damour andR Ruffini ldquoBlack-hole evaporation in the Klein-Sauter-Heisenberg-Euler formalismrdquo Physical Review D vol 14p 332 1976

[5] S M Christensen and S A Fulling ldquoTrace anomalies and theHawking effectrdquo Physical ReviewD vol 15 no 8 pp 2088ndash21041977

[6] JW York ldquoBlack hole in thermal equilibriumwith a scalar fieldthe back-reactionrdquo Physical Review D vol 31 p 755 1985

[7] Z Zhao and J Y Zhu ldquoNernst theorem and Planck absoluteentropy of black holesrdquo Acta Physica Sinica vol 48 no 8 pp1558ndash1564 1999

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

[9] M K Parikh ldquoEnergy conservation and Hawking radiationrdquohttparxivorgabshep-th0402166

[10] K Srinivasan and T Padmanabhan ldquoParticle production andcomplex path analysisrdquo Physical Review D vol 60 no 2 ArticleID 024007 p 20 1999

[11] S Shankaranarayanan K Srinivasan and T PadmanabhanldquoMethod of complex paths and general covariance of HawkingradiationrdquoModern Physics Letters A Particles and Fields Grav-itation Cosmology Nuclear Physics vol 16 no 9 pp 571ndash5782001

[12] R Banerjee B R Majhi and S Samanta ldquoNoncommutativeblack hole thermodynamicsrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 77 no 12 Article ID124035 p 8 2008

Advances in High Energy Physics 9

[13] Q-Q Jiang S-Q Wu and X Cai ldquoHawking radiation as tun-neling from the Kerr and Kerr-Newman black holesrdquo PhysicalReview D vol 73 no 6 Article ID 064003 p 10 2006

[14] Y Hu J Zhang and Z Zhao ldquoMassive particlesrsquo Hawkingradiation via tunneling from the G H dilaton black holerdquoModern Physics Letters A Particles and Fields GravitationCosmology Nuclear Physics vol 21 no 28 pp 2143ndash2149 2006

[15] Z Xu and B Chen ldquoHawking radiation from general Kerr-(anti)de Sitter black holesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 75 no 2 Article ID 024041 p6 2007

[16] R Kerner and R B Mann ldquoFermions tunnelling from blackholesrdquo Classical and Quantum Gravity vol 25 no 9 Article ID095014 p 17 2008

[17] R D Kerner and R B Mann ldquoCharged fermions tunnellingfrom Kerr-Newman black holesrdquo Physics Letters B vol 665 no4 pp 277ndash283 2008

[18] D-Y Chen Q-Q Jiang and X-T Zu ldquoHawking radiation ofDirac particles via tunnelling from rotating black holes in deSitter spacesrdquo Physics Letters B vol 665 no 2-3 pp 106ndash1102008

[19] R Banerjee and B R Majhi ldquoQuantum tunneling beyondsemiclassical approximationrdquo Journal of High Energy Physicsno 6 Article ID 095 p 20 2008

[20] C Corda ldquoEffective temperature for black holesrdquo Journal ofHigh Energy Physics vol 1108 p 101 2011

[21] C Corda S H Hendi R Katebi and N O Schmidt ldquoEffectivestate Hawking radiation and quasi-normal modes for Kerrblack holesrdquo Journal of High Energy Physics no 6 p 008 2013

[22] H M Siahaan and T Triyanta ldquoSemiclassical methods forhawking radiation from a vaidya black holerdquo InternationalJournal of Modern Physics A vol 25 no 1 pp 145ndash153 2010

[23] B D Chowdhury ldquoProblems with tunneling of thin shells fromblack holesrdquo Pramana vol 70 no 1 pp 593ndash612 2008

[24] E T Akhmedov V Akhmedova and D Singleton ldquoHawkingtemperature in the tunneling picturerdquo Physics Letters B vol 642no 1-2 pp 124ndash128 2006

[25] E T Akhmedov T Pilling and D Singleton ldquoSubtleties in thequasi-classical calculation of Hawking radiationrdquo InternationalJournal of Modern Physics D Gravitation Astrophysics Cosmol-ogy vol 17 no 13-14 pp 2453ndash2458 2008

[26] R Banerjee and B R Majhi ldquoQuantum tunnelingand backreactionrdquo Physics Letters B vol 662 no 1 pp 62ndash65 2008

[27] C Corda ldquoEffective temperature Hawking radiation and quasi-normal modesrdquo International Journal of Modern Physics D vol21 no 11 Article ID 1242023 2012

[28] H-L Li and S-Z Yang ldquoHawking radiation from the chargedBTZ black hole with back-reactionrdquo Europhysics Letters vol 79no 2 Article ID 20001 p 3 2007

[29] C Corda S H Hendi R Katebi and N O Schmidt ldquoHawk-ing radiation-quasi-normal modes correspondence and effec-tive states for nonextremal Reissner-Nordstrom black holesrdquoAdvances in High Energy Physics vol 2014 Article ID 5278749 pages 2014

[30] Q-Q Jiang and S-Q Wu ldquoHawking radiation of chargedparticles as tunneling from Reissner-Nordstrom-de Sitter blackholes with a global monopolerdquo Physics Letters B vol 635 no2-3 pp 151ndash155 2006

[31] Y Hu J Zhang and Z Zhao ldquoMassive uncharged and chargedparticlesrsquo tunneling from the Horowitz-Strominger dilatonblack holerdquo International Journal of Modern Physics D Gravi-tation Astrophysics Cosmology vol 16 no 5 pp 847ndash855 2007

[32] C O Lousto and N G Sanchez ldquoBack reaction effects in blackhole spacetimes rdquo Physics Letters B vol 212 no 4 pp 411ndash4141988

[33] P Mitra ldquoHawking temperature from tunnelling formalismrdquoPhysics Letters B vol 648 no 2-3 pp 240ndash242 2007

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

Advances in High Energy Physics 3

From the above expression it is to be noted that 119879119867is time

dependentRecently a drawback of the above approach

has been noted [23ndash25] It has been shown thatΓ sim expminus(2ℏ) Im 119878 = expminus(2ℏ) Imint

119903out

119903in119901119903119889119903 is not

canonically invariant and hence is not a proper observable itshould be modified as expminus Im∮119901

119903119889119903ℏ The closed path

goes across the horizon and back For tunnelling across theordinary barrier it is immaterial whether the particle goesfrom the left to the right or the reverse path So in that case

∮119901119903119889119903 = 2int

119903out

119903in

119901119903119889119903 (12)

and there is no problem of canonical invariance But difficultyarises for BH horizon which behaves as a barrier for particlesgoing from inside of the BH to outside but it does not actas a barrier for particles going from outside to the insideSo relation (12) is no longer valid Also using tunnelling theprobability is Γ sim expminus Im∮119901

119903119889119903ℏ so there will be a

problem of factor two in Hawking temperature [24 26 27]Further the above analysis of tunnelling approach

remains incomplete unless effects of self-gravitation and backreaction are taken into account But unfortunately no generalapproaches to account for the above effects are there in theliterature only few results are available for some known BHsolutions [26ndash32]

Finally it is worth mentioning that so far the abovetunnelling approach is purely semiclassical in nature andquantum corrections are not included Also this method isapplicable for Painleve-type coordinates only one cannot usethe original metric coordinates to avoid horizon singularityLastly the tunnelling approach is not applicable for massiveparticles [19]

3 Hamilton-Jacobi MethodQuantum Corrections

We will now follow the alternative approach as mentioned inthe introduction that is the HJ method to evaluate the imag-inary part of the action and hence the Hawking temperatureWe will analyze the beyond semiclassical approximation byincorporating possible quantum corrections As this methodis not affected by the coordinate singularity at the horizon sowe will use the general BH metric (2) for convenience

In the background of the gravitational field describedby the metric (2) massless scalar particles obey the Klein-Gordon equation

ℏ2

radicminus119892120597 [119892120583]radicminus119892120597]] 120595 = 0 (13)

For spherically symmetric BH as we are only consideringradial trajectories so we will consider (119905 119903)-sector in thespacetime given by (2) that is we concentrate on two-dimensional BH problems Using (2) the above Klein-Gordon equation becomes

1205972120595

1205971199052minus

1

2119860119861

120597 (119860119861)

120597119905

120597120595

120597119905minus1

2

120597 (119860119861)

120597119903

120597120595

120597119903minus 119860119861

1205972120595

1205971199032= 0 (14)

Using the standard ansatz for the semiclassical wave functionnamely

120595 (119903 119905) = exp minus 119894ℏ119878 (119903 119905) (15)

the differential equation for the action 119878 is

(120597119878

120597119905)

2

minus 119860119861(120597119878

120597119903)

2

+ 119894ℏ [1205972119878

1205971199052minus

1

2119860119861

120597 (119860119861)

120597119905

120597119878

120597119905minus1

2

120597 (119860119861)

120597119903

120597119878

120597119903minus 119860119861

1205972119878

1205971199032]

(16)

To solve this partial differential equation we expand theaction 119878 in powers of Planckrsquos constant ℏ as

119878 (119903 119905) = 1198780 (119903 119905) + Σℏ119896119878119896 (119903 119905) (17)

with 119896 being a positive integer Note that in the aboveexpansion terms of the order of Planckrsquos constant and itshigher powers are considered as quantum corrections overthe semiclassical action 119878

0 Now substituting ansatz (17) for 119878

into (16) and equating different powers of ℏ on both sides weobtain the following set of partial differential equations

ℏ0 (120597119878

120597119905)

2

minus 119860119861(120597119878

120597119903)

2

= 0 (18)

ℏ11205971198780

120597119905

1205971198781

120597119905minus 119860119861

1205971198780

120597119903

1205971198781

120597119903

+119894

2[12059721198780

1205971199052minus

1

2119860119861

120597 (119860119861)

120597119905

1205971198780

120597119905

minus1

2

120597 (119860119861)

120597119903

1205971198780

120597119903minus 119860119861

12059721198780

1205971199032] = 0

(19)

ℏ2 (1205971198781

120597119905)

2

+ 21205971198780

120597119905

1205971198782

120597119905minus 119860119861(

1205971198781

120597119903)

2

minus 21198601198611205971198780

120597119903

1205971198782

120597119903

+ 119894 [12059721198781

1205971199052minus

1

2119860119861

120597 (119860119861)

120597119905

1205971198781

120597119905

minus1

2

120597 (119860119861)

120597119903

1205971198781

120597119903minus 119860119861

12059721198781

1205971199032] = 0

(20)

and so onApparently different order partial differential equations

are very complicated but fortunately there will be lot ofsimplifications if in the partial differential equation corre-sponding to ℏ119896 all previous partial differential equationsare used and finally we obtain identical partial differentialequation namely

ℏ119896120597119878119896

120597119905= plusmnradic119860 (119903 119905) 119861 (119903 119905)

120597119878119896

120597119903 (21)

for 119896 = 0 1 2

4 Advances in High Energy Physics

Thus quantum corrections satisfy the same differentialequation as the semiclassical action 119878

0 Hence the solutions

will be very similar To solve 1198780 it is to be noted that due to

nonstatic BHs the metric coefficients are functions of 119903 and119905 and hence standard HJ method cannot be applied somegeneralization is needed We start with a general metric [22]

1198780 (119903 119905) = int

119905

0

1205960(1199051015840) 119889119905 + 119863

0 (119903 119905) (22)

Here 1205960(119905) behaves as the energy of the emitted particle

and the justification of the choice of the integral is that theoutgoing particle should have time-dependent continuumenergy

Now substituting the above ansatz for 1198780(119903 119905) into (18) and

using the radial null geodesic in the usual metric from (2)namely

119889119903

119889119905= plusmnradic119860119861 (23)

we have

1205971198630

120597119903+1205971198630

120597119905

119889119905

119889119903= ∓1205960 (119905)

119889119905

119889119903 (24)

that is

1198891198630

119889119903= ∓

1205960 (119905)

radic119860119861

(25)

which gives

1198630= ∓1205960 (119905) int

119903

0

119889119903

radic119860119861

(26)

Hence the complete semiclassical action takes the form

1198780 (119903 119905) = int

119905

0

1205960(1199051015840) 1198891199051015840∓ 1205960 (119905) int

119903

0

119889119903

radic119860119861

(27)

Here the minus (or +) sign corresponds to absorption (or emis-sion) particle As solution (27) contains an arbitrary time-dependent function 120596

0(119905) so a general solution for 119878

119896can be

written as

119878119896 (119903 119905) = int

119905

0

120596119896(1199051015840) 1198891199051015840∓ 1205960 (119905) int

119903

0

119889119903

radic119860119861

119896 = 1 2 3

(28)

Thus from (15) using solutions (27) and (28) into (17) thewave functions for absorption and emission of scalar particlecan be expressed as

120595emm (119903 119905) = expminus 119894ℏ[(int

119905

0

1205960(1199051015840) 1198891199051015840

+Σ119896ℏ119896int

119905

0

120596119896(1199051015840) 1198891199051015840)

minus (1205960 (119905) + Σ119896ℏ

119896120596119896 (119905)) int

119903

0

119889119903

radic119860119861

]

120595abs (119903 119905) = expminus 119894ℏ[(int

119905

0

1205960(1199051015840) 1198891199051015840

+Σ119896ℏ119896int

119905

0

120596119896(1199051015840) 1198891199051015840)

+ (1205960 (119905) + Σ119896ℏ

119896120596119896 (119905)) int

119903

0

119889119903

radic119860119861

]

(29)

respectively Due to tunnelling across the horizon there willbe a change of sign of the metric coefficients in the (119903 119905)-partof the metric and as a result function of 119905 coordinate has animaginary part which will contribute to the probabilities Sowe write

119875abs =1003816100381610038161003816120595abs (119903 119905)

1003816100381610038161003816

2

= exp2 Imℏ

[(int

119905

0

1205960(1199051015840) 1198891199051015840+ Σ119896ℏ119896int

119905

0

120596119896(1199051015840) 1198891199051015840)

+ (1205960 (119905) + Σ119896ℏ

119896120596119896 (119905)) int

119903

0

119889119903

radic119860119861

]

(30)

119875emm =1003816100381610038161003816120595emm (119903 119905)

1003816100381610038161003816

2

= exp2 Imℏ

[(int

119905

0

1205960(1199051015840) 1198891199051015840+ Σ119896ℏ119896int

119905

0

120596119896(1199051015840) 1198891199051015840)

minus (1205960 (119905) + Σ119896ℏ

119896120596119896 (119905)) int

119903

0

119889119903

radic119860119861

]

(31)

To have some simplification we will now use the physical factthat all incoming particles certainly cross the horizon that is119875abs = 1 So from (30)

Im(int

119905

0

1205960(1199051015840) 1198891199051015840+ Σ119896ℏ119896int

119905

0

120596119896(1199051015840) 1198891199051015840)

= minus Im (1205960 (119905) + Σ119896ℏ

119896120596119896 (119905)) int

119903

0

119889119903

radic119860119861

(32)

and hence 119875emm simplifies to

119875emm = expminus4ℏ(1205960 (119905) + Σ119896ℏ

119896120596119896 (119905)) Imint

119903

0

119889119903

radic119860119861

(33)

Advances in High Energy Physics 5

Then from the principle of ldquodetailed balancerdquo [10ndash12] (whichstates that transitions between any two states take place withequal frequency in either direction at equilibrium) we write

119875emm = expminus1205960 (119905)

119879ℎ

119875in = expminus1205960 (119905)

119879ℎ

(34)

So comparing (33) and (34) the temperature of the BH isgiven by

119879ℎ=ℏ

4[1 + Σ

119896ℏ119896120596119896(119905)

1205960(119905)]

minus1

[Imint

119903

0

119889119903

radic119860119861

]

minus1

(35)

where

119879ℎ=ℏ

4[Imint

119903

0

119889119903

radic119860119861

]

minus1

(36)

is the usual Hawking temperature of the BH Thus due toquantum corrections the temperature of the BH is modifiedfrom the Hawking temperature and both temperatures arefunctions of 119905 and 119903 Note that (36) is the standard expressionfor semiclassical Hawking temperature and it is valid fornonspherical metric also However for spherical metric onecan use the Taylor series expansions (7) near the horizonand obtain 119879

119867as given in (11) by performing the contour

integration The ambiguity of factor of two (as mentionedearlier) in the Hawking temperature does not arise here

Further one may note that solutions (27) or (28) are theunique solutions to (18) or (21) except for a premultiplicationfactor This arbitrary multiplicative factor does not appear inthe expression for Hawking temperature only the particleenergy (120596

0) or 120596

119896is rescaled As quantum correction term

contains1205961198961205960 so it does not involve the arbitrarymultiplica-

tive factor and hence it is uniqueTo have some interpretation about the arbitrary functions

120596119896(119905) appearing in the quantum correction terms we make

use of dimensional analysis As 1198780has the dimension ℏ so the

arbitrary function 120596119896(119905) has the dimension ℏminus119896 In standard

choice of units namely 119866 = 119888 = 119870119861= 1 ℏ sim 119872

2

119901and so

120596119896sim 119872minus2119896 where119872 is the mass of the BH

Similar to the Hawking temperature the surface gravityof the BH is modified due to quantum corrections If 120581

119888is

the semiclassical surface gravity corresponding to Hawkingtemperature that is 120581

119888= 2120587119879

119867 then the quantum corrected

surface gravity 120581 = 2120587119879119867is related to the semiclassical value

by the relation

120581 = 120581119888[1 + Σ

119896ℏ119896120596119896(119905)

1205960(119905)]

minus1

(37)

Moreover based on the dimensional analysis if we choosefor simplicity

120596119896 (119905) =

1198861198961205960 (119905)

1198722119896 ldquo119886rdquo is a dimensionless parameter

(38)

then expression (37) is simplified to

120581 = 1205810(1 minus

ℏ119886

1198722)

minus1

(39)

This is related to the one loop back reaction effects in thespacetime [6 33] with the parameter 119886 corresponding to traceanomaly Higher order loop corrections to the surface gravitycan be obtained similarly by suitable choice of the functions120596119896(119905) For static BHs Banerjee and Majhi [19] have studied

these corrections in detail Lastly it is worth mentioning thatidentical result for BH temperature may be obtained if we usethe Painleve coordinate system as in the previous section

4 Entropy Function and Quantum Correction

We will now examine how the semiclassical Bekenstein-Hawking area law namely 119878BH = (1198604ℏ) (119860 is the area of thehorizon) is modified due to quantum corrections describedin the previous section The first law of the BH mechanicswhich is essentially the energy conservation relation relatedthe change of BHmass (119872) to the change of its entropy (119878BH)electric charge (119876) and angular momentum (119869) as

119889119872 = 119879ℎ119889119878BH + Φ119889119876 + Ω119889119869 (40)

Here Ω is the angular velocity and Φ is the electrostaticpotential So for nonrotating uncharged BHs the entropy hasthe simple form

119878BH = int119889119872

119879ℎ

(41)

or using (35) for 119879ℎ we get

119878BH = int[1 + Σ119896ℏ119896120596119896 (119905)

1205960 (119905)

]119889119872

119879119867

(42)

For choice (38) corresponding to one loop back reactioneffects we have from (42) the quantum corrected BH entropyas

119878BH = int[1 +119886ℏ

119872+1198862ℏ2

1198722+ sdot sdot sdot ]

119889119872

119879119867

(43)

The first term is the usual semiclassical Bekenstein-Hawkingentropy and the subsequent terms are the quantum cor-rections of different order For static BHs Banerjee andMajhi [19] have shown the correction terms of which theleading one gives the standard logarithmic correction On theother hand for nonstatic BHs as the proportionality factorsare time-dependent and arbitrary (see (42)) so the leadingorder correction term may not be logarithmic For futurework we will attempt to determine physical interpretation ofthe arbitrary time-dependent proportionality factors so thatquantum corrections may be evaluated

5 Hamilton-Jacobi Method for MassiveParticles Quantum Corrections

The KG equation for a scalar field 120595 describing a scalarparticle of mass119898

0has the form [10]

(◻ +1198982

0

ℏ2)120595 = 0 (44)

6 Advances in High Energy Physics

where the box operator ◻ is evaluated in the background ofa general static BH metric of the form

1198891199042= minus119860 (119903) 119889119905

2+1198891199032

119861 (119903)+ 1199032119889Ω2

2 (45)

The explicit form of the KG equation for the metric (45) is

minus1

119860

1205972120595

1205971199052+ 119861

1205972120595

1205971199032+1

2119860

120597 (119860119861)

120597119903

120597120595

120597119903+2119861

119903

120597120595

120597119903

+1

1199032 sin 120579120597

120597120579(sin 120579

120597120595

120597120579)

+1

1199032sin21205791205972120595

1205971206012=1198982

0

ℏ2120595 (119905 119903 120579 120601)

(46)

Due to spherical symmetry we can decompose 120601 in the form

120595 (119905 119903 120579 120601) = Φ (119905 119903) 119884119898

119897(120579 120601) (47)

where 120601 satisfies [10]

1

119860

1205972120595

1205971199052minus 119861

1205972120595

1205971199032minus1

2119860

120597 (119860119861)

120597119903

120597120595

120597119903minus2119861

119903

120597120595

120597119903

+ 119897 (119897 + 1)

1199032+1198982

0

ℏ2Φ (119905 119903) = 0

(48)

If we substitute the standard ansatz for the semiclassical wavefunction namely

120601 (119905 119903) = exp minus 119894ℏ119878 (119903 119905) (49)

then the action 119878 will satisfy the following differential equa-tion

[1

119860(120597119878

120597119905)

2

minus 119861(120597119878

120597119903)

2

minus 1198642

0(119903)]

minusℏ

119894[1

119860

1205972119878

1205971199052minus 1198612 1205972119878

1205971199032minus

1

2119860

120597 (119860119861)

120597119903+2119861

119903120597119878

120597119903] = 0

(50)

where 11986420= 1198982

0+ (11987121199032) and 1198712 = 119897(119897 + 1)ℏ2 is the angular

momentum To incorporate quantum corrections over thesemiclassical action we expand the actions in powers ofPlanck constant ℏ as

119878 (119903 119905) = 1198780 (119903 119905) + Σ119896ℏ119896119878119896 (119903 119905) (51)

where 1198780is the semiclassical action and 119896 is a positive integer

Now substituting this ansatz for 119878 in the differential equation

(50) and equating different powers of ℏ on both sides weobtain the following set of partial differential equations

ℏ01

119860(120597119878

120597119905)

2

minus 119861(120597119878

120597119903)

2

minus 1198642

0(119903) = 0 (52)

ℏ12

119860

1205971198780

120597119905

1205971198781

120597119905minus 2119861

1205971198780

120597119903

1205971198781

120597119903

minus1

119894[1

119860

12059721198780

1205971199052minus 1198612 12059721198780

1205971199032

minus1

2119860

120597 (119860119861)

120597119903+2119861

1199031205971198780

120597119903] = 0

(53)

ℏ21

119860(1205971198781

120597119905)

2

+2

119860

1205971198780

120597119905

1205971198782

120597119905minus 119861(

1205971198781

120597119903)

2

minus 21198611205971198780

120597119903

1205971198782

120597119903

minus1

119894[1

119860

12059721198781

1205971199052minus 1198612 12059721198781

1205971199032

minus1

2119860

120597 (119860119861)

120597119903+2119861

1199031205971198781

120597119903] = 0

(54)

and so onTo solve the semiclassical action 119878

0 we start with the

standard separable choice [10]

1198780 (119903 119905) = 1205960119905 + 1198630 (119903) (55)

Substituting this choice in (52) we obtain

1198630= plusmnint

119903

0

radic1205962

0minus 1198601198642

0

119860119861119889119903 = plusmn119868

0(say) (56)

where + or minus sign corresponds to absorption or emission ofscalar particle Now substituting this choice for 119878

0in (53) we

have the differential equation for first order corrections 1198781as

1205971198781

120597119905∓ radic119860119861radic1 minus

1198601198642

0

1205962

0

1205971198781

120597119903

∓radic119860119861

119894

[[

[

minus1

119903

radic1 minus1198601198642

0

1205962

+

(120597119860120597119903) (1198642

01205962) minus (2119860119871

21205962

01199033)

4radic1 minus (1198601198642

01205962)

]]

]

= 0

(57)

As before 1198781can be written in separable form as

1198781= 1205961119905 + 1198631 (119903) (58)

Advances in High Energy Physics 7

where

1198631= int

119903

0

119889119903

radic119860119861radic1 minus (1198601198642

01205962

0)

times[[

[

plusmn 1205961minusradic119860119861

119894

times

minus1

119903

radic1 minus1198601198642

0

1205962

+

(120597119860120597119903) (1198642

01205962) minus (2119860119871

21205962

01199033)

4radic1 minus (1198601198642

01205962)

]]

]

= plusmn1198681minus 1198682

(59)

Now due to complicated form if we retain terms up to firstorder quantum corrections that is

119878 = 1198780+ ℏ1198781= (1205960+ ℏ1205961) 119905 + 119863

0+ ℏ1198631 (119903) (60)

then the wave function denoting absorption and emissionsolutions of the KG equation (48) using (49) are of the form

120601abs = exp minus 119894ℏ(1205960+ ℏ1205961119905 + 1198680+ ℏ1198681minus ℏ1198682)

120601emm = exp minus 119894ℏ(1205960+ ℏ1205961119905 minus 1198680+ ℏ1198681minus ℏ1198682)

(61)

It is to be noted that in course of tunnelling across the hori-zon the coordinate nature changes that is more preciselythe signs of themetric coefficients in the (119903 119905)-hyperplane arealteredThus we can interpret this as that the time coordinatehas an imaginary part in crossing the horizon and accordinglythe temporal part has contribution to the probabilities [1933] Thus absorption and emission probabilities are given by

119875abs =1003816100381610038161003816120601in

1003816100381610038161003816

2= exp 2

ℏ(Im 120596

0+ ℏ1205961119905)

+ Im 1198680+ ℏ1198681minus Im ℏ119868

2

(62)

119875emm =1003816100381610038161003816120601out

1003816100381610038161003816

2= exp minus 119894

ℏ(Im 120596

0+ ℏ1205961119905)

minus Im 1198680+ ℏ1198681minus Im ℏ119868

2

(63)

In the classical limit ℏ rarr 0 there is no reflection so allingoing particles should be absorbed and hence [33]

limℏrarr0

119875abs = 1 (64)

So from (62) we must have

Im 1205960119905 = Im 119868

0 Im (120596

1119905 minus 1198682) = Im 119868

1 (65)

and as a result 119875emm simplifies to

119875emm = exp[[

[

minus41205960

ℏ

times Im

int

119903

0

119889119903

radic119860119861

(radic1 minus1198601198642

0

1205962

0

+ℏ (12059611205960)

radic1 minus (1198601198642

01205962

0)

)

]]

]

(66)

Using the principle of ldquodetailed balancerdquo [10 11 20 21]namely

119875emm = expminus 119864119879ℎ

119875in = expminus 119864119879ℎ

(67)

the temperature of the BH is given by

119879ℎ=ℏ119864

41205960

[[

[

Im

int

119903

0

119889119903

radic119860119861

times (radic1 minus1198601198642

0

1205962

0

+ℏ(12059611205960)

radic1 minus (1198601198642

01205962

0)

)

]]

]

minus1

(68)

where the semiclassical Hawking temperature of the BH hasthe expression

119879119867=ℏ119864

41205960

[

[

Imint

119903

0

119889119903

radic119860119861

radic1 minus1198601198642

0

1205962

0

]

]

minus1

(69)

Now to obtain themodified form of the surface gravity of theBH we start with the usual relation between surface gravityand Hawking temperature namely

120581119867= 2120587119879

119867 (70)

where 119879119867is given by (69)

So the quantum corrected surface gravity is given by

120581QC = 2120587119879ℎ (71)

Further for the present nonrotating uncharged static BHsusing the law of BH thermodynamics 119889119872 = 119879

ℎ119889119878 we have

the expression for the entropy of the BH as

119878BH = int41205960

ℏ119864(1 +

ℏ1205961

1205960

)119889119872int

119903

0

119889119903

radic119860119861

(72)

Finally it is easy to see from (68) that near the horizonthe presence of 1198642

0term can be neglected as it is multiplied

8 Advances in High Energy Physics

by themetric coefficient119860Therefore the quantum corrected(up to first order) temperature of the BH (in (68)) reduces to

119879ℎ=ℏ119864

41205960

(1 +ℏ1205961

1205960

)

minus1

[int

119903

0

119889119903

radic119860119861

]

minus1

(73)

and the Hawking temperature (given in (69)) becomes

119879119867=ℏ119864

41205960

[int

119903

0

119889119903

radic119860119861

]

minus1

(74)

So we have

119879ℎ= (1 +

ℏ1205961

1205960

)

minus1

119879119867 (75)

We see that if the energy of the tunnelling particle is chosen as1205960(ie 119864 = 120596

0) and 120596

1= 1205731119872 (for notations see Banerjee

and Majhi [19]) then the Hawking temperature given by(74) is the usual one derived for massless particles andthe quantum corrected temperature 119879

ℎgiven in (75) agrees

with that of Banerjee and Majhi [19] for massless particleTherefore Hawking temperature near the horizon remainsthe same for both massless and nonzero mass tunnellingparticles and it agrees with the claim of Srinivasan andPadmanabhan [10] and Banerjee and Majhi [19] For futurework it will be interesting to calculate the temperature of theBH for tunnelling nonzero mass particle with full quantumcorrections and examine whether the result agrees with thatof Banerjee andMajhi [19] near the horizon Finally it will benice to determine quantum corrected entropy of the BH in aconvenient form

6 Summary of the Work

This work is an attempt to study quantum corrections toHawking radiation of massless particle from a dynamical BHas well as for massive particle from a static BH At firstradial null geodesic tunnelling approach has been used withPainleve-type choice of coordinate system to derive semiclas-sical Hawking temperature Then full quantum mechanicalcalculations have performed writing action in a power seriesof the Planck constant ℏ to evaluate the quantum correc-tions to the Hawking temperature Subsequently quantumcorrected surface gravity has been calculated and it is foundthat one loop back reaction effects in the spacetime can beobtained by suitable choice of the arbitrary functions andparameters Finally an expression for the quantum correctedentropy of the BH has been evaluated It is found that dueto the presence of the arbitrary functions in the expressionfor entropy the leading order quantum correction may notbe logarithmic in nature On the other hand in the caseof Hawking radiation of massive particle from static BH itis found that Hawking temperature near the horizon doesnot depend on the mass term as predicted by Srinivasanand Padmanabhan [10] and Banerjee et al [16ndash18] Forfuture work we will try to find a solution for the partialdifferential equation (18) in a more simple form so that morephysical interpretations can be done from the BHparameters

Also it will be interesting to calculate temperature of theBH for tunnelling nonzero mass particle with full quantumcorrection and examine whether the result agrees with thatof Banerjee andMajhi [19] near the horizon Finally it will benice to determine quantum corrected entropy of the BH in aconvenient form

Conflict of Interests

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

Acknowledgments

The authors are thankful to IUCAA Pune India for theirwarm hospitality and research facilities as the work hasbeen done there during a visit Also Subenoy Chakrabortyacknowledges the UGC-DRS Programme in the Departmentof Mathematics Jadavpur University The author SubhajitSaha is thankful to UGC-BSR Programme of JadavpurUniversity for awarding research fellowship The authors arethankful to Ritabrata Biswas and Nairwita Mazumder fortheir help at the initial part of the work

References

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

[2] J B Hartle and S W Hawking ldquoPath-integral derivation ofblack-hole radiancerdquo Physical Review D vol 13 no 8 pp 2188ndash2203 1976

[3] J D Bekenstein ldquoGeneralized second law of thermodynamicsin black-hole physicsrdquo Physical Review D vol 9 p 2188 1974

[4] T Damour andR Ruffini ldquoBlack-hole evaporation in the Klein-Sauter-Heisenberg-Euler formalismrdquo Physical Review D vol 14p 332 1976

[5] S M Christensen and S A Fulling ldquoTrace anomalies and theHawking effectrdquo Physical ReviewD vol 15 no 8 pp 2088ndash21041977

[6] JW York ldquoBlack hole in thermal equilibriumwith a scalar fieldthe back-reactionrdquo Physical Review D vol 31 p 755 1985

[7] Z Zhao and J Y Zhu ldquoNernst theorem and Planck absoluteentropy of black holesrdquo Acta Physica Sinica vol 48 no 8 pp1558ndash1564 1999

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

[9] M K Parikh ldquoEnergy conservation and Hawking radiationrdquohttparxivorgabshep-th0402166

[10] K Srinivasan and T Padmanabhan ldquoParticle production andcomplex path analysisrdquo Physical Review D vol 60 no 2 ArticleID 024007 p 20 1999

[11] S Shankaranarayanan K Srinivasan and T PadmanabhanldquoMethod of complex paths and general covariance of HawkingradiationrdquoModern Physics Letters A Particles and Fields Grav-itation Cosmology Nuclear Physics vol 16 no 9 pp 571ndash5782001

[12] R Banerjee B R Majhi and S Samanta ldquoNoncommutativeblack hole thermodynamicsrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 77 no 12 Article ID124035 p 8 2008

Advances in High Energy Physics 9

[13] Q-Q Jiang S-Q Wu and X Cai ldquoHawking radiation as tun-neling from the Kerr and Kerr-Newman black holesrdquo PhysicalReview D vol 73 no 6 Article ID 064003 p 10 2006

[14] Y Hu J Zhang and Z Zhao ldquoMassive particlesrsquo Hawkingradiation via tunneling from the G H dilaton black holerdquoModern Physics Letters A Particles and Fields GravitationCosmology Nuclear Physics vol 21 no 28 pp 2143ndash2149 2006

[15] Z Xu and B Chen ldquoHawking radiation from general Kerr-(anti)de Sitter black holesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 75 no 2 Article ID 024041 p6 2007

[16] R Kerner and R B Mann ldquoFermions tunnelling from blackholesrdquo Classical and Quantum Gravity vol 25 no 9 Article ID095014 p 17 2008

[17] R D Kerner and R B Mann ldquoCharged fermions tunnellingfrom Kerr-Newman black holesrdquo Physics Letters B vol 665 no4 pp 277ndash283 2008

[18] D-Y Chen Q-Q Jiang and X-T Zu ldquoHawking radiation ofDirac particles via tunnelling from rotating black holes in deSitter spacesrdquo Physics Letters B vol 665 no 2-3 pp 106ndash1102008

[19] R Banerjee and B R Majhi ldquoQuantum tunneling beyondsemiclassical approximationrdquo Journal of High Energy Physicsno 6 Article ID 095 p 20 2008

[20] C Corda ldquoEffective temperature for black holesrdquo Journal ofHigh Energy Physics vol 1108 p 101 2011

[21] C Corda S H Hendi R Katebi and N O Schmidt ldquoEffectivestate Hawking radiation and quasi-normal modes for Kerrblack holesrdquo Journal of High Energy Physics no 6 p 008 2013

[22] H M Siahaan and T Triyanta ldquoSemiclassical methods forhawking radiation from a vaidya black holerdquo InternationalJournal of Modern Physics A vol 25 no 1 pp 145ndash153 2010

[23] B D Chowdhury ldquoProblems with tunneling of thin shells fromblack holesrdquo Pramana vol 70 no 1 pp 593ndash612 2008

[24] E T Akhmedov V Akhmedova and D Singleton ldquoHawkingtemperature in the tunneling picturerdquo Physics Letters B vol 642no 1-2 pp 124ndash128 2006

[25] E T Akhmedov T Pilling and D Singleton ldquoSubtleties in thequasi-classical calculation of Hawking radiationrdquo InternationalJournal of Modern Physics D Gravitation Astrophysics Cosmol-ogy vol 17 no 13-14 pp 2453ndash2458 2008

[26] R Banerjee and B R Majhi ldquoQuantum tunnelingand backreactionrdquo Physics Letters B vol 662 no 1 pp 62ndash65 2008

[27] C Corda ldquoEffective temperature Hawking radiation and quasi-normal modesrdquo International Journal of Modern Physics D vol21 no 11 Article ID 1242023 2012

[28] H-L Li and S-Z Yang ldquoHawking radiation from the chargedBTZ black hole with back-reactionrdquo Europhysics Letters vol 79no 2 Article ID 20001 p 3 2007

[29] C Corda S H Hendi R Katebi and N O Schmidt ldquoHawk-ing radiation-quasi-normal modes correspondence and effec-tive states for nonextremal Reissner-Nordstrom black holesrdquoAdvances in High Energy Physics vol 2014 Article ID 5278749 pages 2014

[30] Q-Q Jiang and S-Q Wu ldquoHawking radiation of chargedparticles as tunneling from Reissner-Nordstrom-de Sitter blackholes with a global monopolerdquo Physics Letters B vol 635 no2-3 pp 151ndash155 2006

[31] Y Hu J Zhang and Z Zhao ldquoMassive uncharged and chargedparticlesrsquo tunneling from the Horowitz-Strominger dilatonblack holerdquo International Journal of Modern Physics D Gravi-tation Astrophysics Cosmology vol 16 no 5 pp 847ndash855 2007

[32] C O Lousto and N G Sanchez ldquoBack reaction effects in blackhole spacetimes rdquo Physics Letters B vol 212 no 4 pp 411ndash4141988

[33] P Mitra ldquoHawking temperature from tunnelling formalismrdquoPhysics Letters B vol 648 no 2-3 pp 240ndash242 2007

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

4 Advances in High Energy Physics

Thus quantum corrections satisfy the same differentialequation as the semiclassical action 119878

0 Hence the solutions

will be very similar To solve 1198780 it is to be noted that due to

nonstatic BHs the metric coefficients are functions of 119903 and119905 and hence standard HJ method cannot be applied somegeneralization is needed We start with a general metric [22]

1198780 (119903 119905) = int

119905

0

1205960(1199051015840) 119889119905 + 119863

0 (119903 119905) (22)

Here 1205960(119905) behaves as the energy of the emitted particle

and the justification of the choice of the integral is that theoutgoing particle should have time-dependent continuumenergy

Now substituting the above ansatz for 1198780(119903 119905) into (18) and

using the radial null geodesic in the usual metric from (2)namely

119889119903

119889119905= plusmnradic119860119861 (23)

we have

1205971198630

120597119903+1205971198630

120597119905

119889119905

119889119903= ∓1205960 (119905)

119889119905

119889119903 (24)

that is

1198891198630

119889119903= ∓

1205960 (119905)

radic119860119861

(25)

which gives

1198630= ∓1205960 (119905) int

119903

0

119889119903

radic119860119861

(26)

Hence the complete semiclassical action takes the form

1198780 (119903 119905) = int

119905

0

1205960(1199051015840) 1198891199051015840∓ 1205960 (119905) int

119903

0

119889119903

radic119860119861

(27)

Here the minus (or +) sign corresponds to absorption (or emis-sion) particle As solution (27) contains an arbitrary time-dependent function 120596

0(119905) so a general solution for 119878

119896can be

written as

119878119896 (119903 119905) = int

119905

0

120596119896(1199051015840) 1198891199051015840∓ 1205960 (119905) int

119903

0

119889119903

radic119860119861

119896 = 1 2 3

(28)

Thus from (15) using solutions (27) and (28) into (17) thewave functions for absorption and emission of scalar particlecan be expressed as

120595emm (119903 119905) = expminus 119894ℏ[(int

119905

0

1205960(1199051015840) 1198891199051015840

+Σ119896ℏ119896int

119905

0

120596119896(1199051015840) 1198891199051015840)

minus (1205960 (119905) + Σ119896ℏ

119896120596119896 (119905)) int

119903

0

119889119903

radic119860119861

]

120595abs (119903 119905) = expminus 119894ℏ[(int

119905

0

1205960(1199051015840) 1198891199051015840

+Σ119896ℏ119896int

119905

0

120596119896(1199051015840) 1198891199051015840)

+ (1205960 (119905) + Σ119896ℏ

119896120596119896 (119905)) int

119903

0

119889119903

radic119860119861

]

(29)

respectively Due to tunnelling across the horizon there willbe a change of sign of the metric coefficients in the (119903 119905)-partof the metric and as a result function of 119905 coordinate has animaginary part which will contribute to the probabilities Sowe write

119875abs =1003816100381610038161003816120595abs (119903 119905)

1003816100381610038161003816

2

= exp2 Imℏ

[(int

119905

0

1205960(1199051015840) 1198891199051015840+ Σ119896ℏ119896int

119905

0

120596119896(1199051015840) 1198891199051015840)

+ (1205960 (119905) + Σ119896ℏ

119896120596119896 (119905)) int

119903

0

119889119903

radic119860119861

]

(30)

119875emm =1003816100381610038161003816120595emm (119903 119905)

1003816100381610038161003816

2

= exp2 Imℏ

[(int

119905

0

1205960(1199051015840) 1198891199051015840+ Σ119896ℏ119896int

119905

0

120596119896(1199051015840) 1198891199051015840)

minus (1205960 (119905) + Σ119896ℏ

119896120596119896 (119905)) int

119903

0

119889119903

radic119860119861

]

(31)

To have some simplification we will now use the physical factthat all incoming particles certainly cross the horizon that is119875abs = 1 So from (30)

Im(int

119905

0

1205960(1199051015840) 1198891199051015840+ Σ119896ℏ119896int

119905

0

120596119896(1199051015840) 1198891199051015840)

= minus Im (1205960 (119905) + Σ119896ℏ

119896120596119896 (119905)) int

119903

0

119889119903

radic119860119861

(32)

and hence 119875emm simplifies to

119875emm = expminus4ℏ(1205960 (119905) + Σ119896ℏ

119896120596119896 (119905)) Imint

119903

0

119889119903

radic119860119861

(33)

Advances in High Energy Physics 5

Then from the principle of ldquodetailed balancerdquo [10ndash12] (whichstates that transitions between any two states take place withequal frequency in either direction at equilibrium) we write

119875emm = expminus1205960 (119905)

119879ℎ

119875in = expminus1205960 (119905)

119879ℎ

(34)

So comparing (33) and (34) the temperature of the BH isgiven by

119879ℎ=ℏ

4[1 + Σ

119896ℏ119896120596119896(119905)

1205960(119905)]

minus1

[Imint

119903

0

119889119903

radic119860119861

]

minus1

(35)

where

119879ℎ=ℏ

4[Imint

119903

0

119889119903

radic119860119861

]

minus1

(36)

is the usual Hawking temperature of the BH Thus due toquantum corrections the temperature of the BH is modifiedfrom the Hawking temperature and both temperatures arefunctions of 119905 and 119903 Note that (36) is the standard expressionfor semiclassical Hawking temperature and it is valid fornonspherical metric also However for spherical metric onecan use the Taylor series expansions (7) near the horizonand obtain 119879

119867as given in (11) by performing the contour

integration The ambiguity of factor of two (as mentionedearlier) in the Hawking temperature does not arise here

Further one may note that solutions (27) or (28) are theunique solutions to (18) or (21) except for a premultiplicationfactor This arbitrary multiplicative factor does not appear inthe expression for Hawking temperature only the particleenergy (120596

0) or 120596

119896is rescaled As quantum correction term

contains1205961198961205960 so it does not involve the arbitrarymultiplica-

tive factor and hence it is uniqueTo have some interpretation about the arbitrary functions

120596119896(119905) appearing in the quantum correction terms we make

use of dimensional analysis As 1198780has the dimension ℏ so the

arbitrary function 120596119896(119905) has the dimension ℏminus119896 In standard

choice of units namely 119866 = 119888 = 119870119861= 1 ℏ sim 119872

2

119901and so

120596119896sim 119872minus2119896 where119872 is the mass of the BH

Similar to the Hawking temperature the surface gravityof the BH is modified due to quantum corrections If 120581

119888is

the semiclassical surface gravity corresponding to Hawkingtemperature that is 120581

119888= 2120587119879

119867 then the quantum corrected

surface gravity 120581 = 2120587119879119867is related to the semiclassical value

by the relation

120581 = 120581119888[1 + Σ

119896ℏ119896120596119896(119905)

1205960(119905)]

minus1

(37)

Moreover based on the dimensional analysis if we choosefor simplicity

120596119896 (119905) =

1198861198961205960 (119905)

1198722119896 ldquo119886rdquo is a dimensionless parameter

(38)

then expression (37) is simplified to

120581 = 1205810(1 minus

ℏ119886

1198722)

minus1

(39)

This is related to the one loop back reaction effects in thespacetime [6 33] with the parameter 119886 corresponding to traceanomaly Higher order loop corrections to the surface gravitycan be obtained similarly by suitable choice of the functions120596119896(119905) For static BHs Banerjee and Majhi [19] have studied

these corrections in detail Lastly it is worth mentioning thatidentical result for BH temperature may be obtained if we usethe Painleve coordinate system as in the previous section

4 Entropy Function and Quantum Correction

We will now examine how the semiclassical Bekenstein-Hawking area law namely 119878BH = (1198604ℏ) (119860 is the area of thehorizon) is modified due to quantum corrections describedin the previous section The first law of the BH mechanicswhich is essentially the energy conservation relation relatedthe change of BHmass (119872) to the change of its entropy (119878BH)electric charge (119876) and angular momentum (119869) as

119889119872 = 119879ℎ119889119878BH + Φ119889119876 + Ω119889119869 (40)

Here Ω is the angular velocity and Φ is the electrostaticpotential So for nonrotating uncharged BHs the entropy hasthe simple form

119878BH = int119889119872

119879ℎ

(41)

or using (35) for 119879ℎ we get

119878BH = int[1 + Σ119896ℏ119896120596119896 (119905)

1205960 (119905)

]119889119872

119879119867

(42)

For choice (38) corresponding to one loop back reactioneffects we have from (42) the quantum corrected BH entropyas

119878BH = int[1 +119886ℏ

119872+1198862ℏ2

1198722+ sdot sdot sdot ]

119889119872

119879119867

(43)

The first term is the usual semiclassical Bekenstein-Hawkingentropy and the subsequent terms are the quantum cor-rections of different order For static BHs Banerjee andMajhi [19] have shown the correction terms of which theleading one gives the standard logarithmic correction On theother hand for nonstatic BHs as the proportionality factorsare time-dependent and arbitrary (see (42)) so the leadingorder correction term may not be logarithmic For futurework we will attempt to determine physical interpretation ofthe arbitrary time-dependent proportionality factors so thatquantum corrections may be evaluated

5 Hamilton-Jacobi Method for MassiveParticles Quantum Corrections

The KG equation for a scalar field 120595 describing a scalarparticle of mass119898

0has the form [10]

(◻ +1198982

0

ℏ2)120595 = 0 (44)

6 Advances in High Energy Physics

where the box operator ◻ is evaluated in the background ofa general static BH metric of the form

1198891199042= minus119860 (119903) 119889119905

2+1198891199032

119861 (119903)+ 1199032119889Ω2

2 (45)

The explicit form of the KG equation for the metric (45) is

minus1

119860

1205972120595

1205971199052+ 119861

1205972120595

1205971199032+1

2119860

120597 (119860119861)

120597119903

120597120595

120597119903+2119861

119903

120597120595

120597119903

+1

1199032 sin 120579120597

120597120579(sin 120579

120597120595

120597120579)

+1

1199032sin21205791205972120595

1205971206012=1198982

0

ℏ2120595 (119905 119903 120579 120601)

(46)

Due to spherical symmetry we can decompose 120601 in the form

120595 (119905 119903 120579 120601) = Φ (119905 119903) 119884119898

119897(120579 120601) (47)

where 120601 satisfies [10]

1

119860

1205972120595

1205971199052minus 119861

1205972120595

1205971199032minus1

2119860

120597 (119860119861)

120597119903

120597120595

120597119903minus2119861

119903

120597120595

120597119903

+ 119897 (119897 + 1)

1199032+1198982

0

ℏ2Φ (119905 119903) = 0

(48)

If we substitute the standard ansatz for the semiclassical wavefunction namely

120601 (119905 119903) = exp minus 119894ℏ119878 (119903 119905) (49)

then the action 119878 will satisfy the following differential equa-tion

[1

119860(120597119878

120597119905)

2

minus 119861(120597119878

120597119903)

2

minus 1198642

0(119903)]

minusℏ

119894[1

119860

1205972119878

1205971199052minus 1198612 1205972119878

1205971199032minus

1

2119860

120597 (119860119861)

120597119903+2119861

119903120597119878

120597119903] = 0

(50)

where 11986420= 1198982

0+ (11987121199032) and 1198712 = 119897(119897 + 1)ℏ2 is the angular

momentum To incorporate quantum corrections over thesemiclassical action we expand the actions in powers ofPlanck constant ℏ as

119878 (119903 119905) = 1198780 (119903 119905) + Σ119896ℏ119896119878119896 (119903 119905) (51)

where 1198780is the semiclassical action and 119896 is a positive integer

Now substituting this ansatz for 119878 in the differential equation

(50) and equating different powers of ℏ on both sides weobtain the following set of partial differential equations

ℏ01

119860(120597119878

120597119905)

2

minus 119861(120597119878

120597119903)

2

minus 1198642

0(119903) = 0 (52)

ℏ12

119860

1205971198780

120597119905

1205971198781

120597119905minus 2119861

1205971198780

120597119903

1205971198781

120597119903

minus1

119894[1

119860

12059721198780

1205971199052minus 1198612 12059721198780

1205971199032

minus1

2119860

120597 (119860119861)

120597119903+2119861

1199031205971198780

120597119903] = 0

(53)

ℏ21

119860(1205971198781

120597119905)

2

+2

119860

1205971198780

120597119905

1205971198782

120597119905minus 119861(

1205971198781

120597119903)

2

minus 21198611205971198780

120597119903

1205971198782

120597119903

minus1

119894[1

119860

12059721198781

1205971199052minus 1198612 12059721198781

1205971199032

minus1

2119860

120597 (119860119861)

120597119903+2119861

1199031205971198781

120597119903] = 0

(54)

and so onTo solve the semiclassical action 119878

0 we start with the

standard separable choice [10]

1198780 (119903 119905) = 1205960119905 + 1198630 (119903) (55)

Substituting this choice in (52) we obtain

1198630= plusmnint

119903

0

radic1205962

0minus 1198601198642

0

119860119861119889119903 = plusmn119868

0(say) (56)

where + or minus sign corresponds to absorption or emission ofscalar particle Now substituting this choice for 119878

0in (53) we

have the differential equation for first order corrections 1198781as

1205971198781

120597119905∓ radic119860119861radic1 minus

1198601198642

0

1205962

0

1205971198781

120597119903

∓radic119860119861

119894

[[

[

minus1

119903

radic1 minus1198601198642

0

1205962

+

(120597119860120597119903) (1198642

01205962) minus (2119860119871

21205962

01199033)

4radic1 minus (1198601198642

01205962)

]]

]

= 0

(57)

As before 1198781can be written in separable form as

1198781= 1205961119905 + 1198631 (119903) (58)

Advances in High Energy Physics 7

where

1198631= int

119903

0

119889119903

radic119860119861radic1 minus (1198601198642

01205962

0)

times[[

[

plusmn 1205961minusradic119860119861

119894

times

minus1

119903

radic1 minus1198601198642

0

1205962

+

(120597119860120597119903) (1198642

01205962) minus (2119860119871

21205962

01199033)

4radic1 minus (1198601198642

01205962)

]]

]

= plusmn1198681minus 1198682

(59)

Now due to complicated form if we retain terms up to firstorder quantum corrections that is

119878 = 1198780+ ℏ1198781= (1205960+ ℏ1205961) 119905 + 119863

0+ ℏ1198631 (119903) (60)

then the wave function denoting absorption and emissionsolutions of the KG equation (48) using (49) are of the form

120601abs = exp minus 119894ℏ(1205960+ ℏ1205961119905 + 1198680+ ℏ1198681minus ℏ1198682)

120601emm = exp minus 119894ℏ(1205960+ ℏ1205961119905 minus 1198680+ ℏ1198681minus ℏ1198682)

(61)

It is to be noted that in course of tunnelling across the hori-zon the coordinate nature changes that is more preciselythe signs of themetric coefficients in the (119903 119905)-hyperplane arealteredThus we can interpret this as that the time coordinatehas an imaginary part in crossing the horizon and accordinglythe temporal part has contribution to the probabilities [1933] Thus absorption and emission probabilities are given by

119875abs =1003816100381610038161003816120601in

1003816100381610038161003816

2= exp 2

ℏ(Im 120596

0+ ℏ1205961119905)

+ Im 1198680+ ℏ1198681minus Im ℏ119868

2

(62)

119875emm =1003816100381610038161003816120601out

1003816100381610038161003816

2= exp minus 119894

ℏ(Im 120596

0+ ℏ1205961119905)

minus Im 1198680+ ℏ1198681minus Im ℏ119868

2

(63)

In the classical limit ℏ rarr 0 there is no reflection so allingoing particles should be absorbed and hence [33]

limℏrarr0

119875abs = 1 (64)

So from (62) we must have

Im 1205960119905 = Im 119868

0 Im (120596

1119905 minus 1198682) = Im 119868

1 (65)

and as a result 119875emm simplifies to

119875emm = exp[[

[

minus41205960

ℏ

times Im

int

119903

0

119889119903

radic119860119861

(radic1 minus1198601198642

0

1205962

0

+ℏ (12059611205960)

radic1 minus (1198601198642

01205962

0)

)

]]

]

(66)

Using the principle of ldquodetailed balancerdquo [10 11 20 21]namely

119875emm = expminus 119864119879ℎ

119875in = expminus 119864119879ℎ

(67)

the temperature of the BH is given by

119879ℎ=ℏ119864

41205960

[[

[

Im

int

119903

0

119889119903

radic119860119861

times (radic1 minus1198601198642

0

1205962

0

+ℏ(12059611205960)

radic1 minus (1198601198642

01205962

0)

)

]]

]

minus1

(68)

where the semiclassical Hawking temperature of the BH hasthe expression

119879119867=ℏ119864

41205960

[

[

Imint

119903

0

119889119903

radic119860119861

radic1 minus1198601198642

0

1205962

0

]

]

minus1

(69)

Now to obtain themodified form of the surface gravity of theBH we start with the usual relation between surface gravityand Hawking temperature namely

120581119867= 2120587119879

119867 (70)

where 119879119867is given by (69)

So the quantum corrected surface gravity is given by

120581QC = 2120587119879ℎ (71)

Further for the present nonrotating uncharged static BHsusing the law of BH thermodynamics 119889119872 = 119879

ℎ119889119878 we have

the expression for the entropy of the BH as

119878BH = int41205960

ℏ119864(1 +

ℏ1205961

1205960

)119889119872int

119903

0

119889119903

radic119860119861

(72)

Finally it is easy to see from (68) that near the horizonthe presence of 1198642

0term can be neglected as it is multiplied

8 Advances in High Energy Physics

by themetric coefficient119860Therefore the quantum corrected(up to first order) temperature of the BH (in (68)) reduces to

119879ℎ=ℏ119864

41205960

(1 +ℏ1205961

1205960

)

minus1

[int

119903

0

119889119903

radic119860119861

]

minus1

(73)

and the Hawking temperature (given in (69)) becomes

119879119867=ℏ119864

41205960

[int

119903

0

119889119903

radic119860119861

]

minus1

(74)

So we have

119879ℎ= (1 +

ℏ1205961

1205960

)

minus1

119879119867 (75)

We see that if the energy of the tunnelling particle is chosen as1205960(ie 119864 = 120596

0) and 120596

1= 1205731119872 (for notations see Banerjee

and Majhi [19]) then the Hawking temperature given by(74) is the usual one derived for massless particles andthe quantum corrected temperature 119879

ℎgiven in (75) agrees

with that of Banerjee and Majhi [19] for massless particleTherefore Hawking temperature near the horizon remainsthe same for both massless and nonzero mass tunnellingparticles and it agrees with the claim of Srinivasan andPadmanabhan [10] and Banerjee and Majhi [19] For futurework it will be interesting to calculate the temperature of theBH for tunnelling nonzero mass particle with full quantumcorrections and examine whether the result agrees with thatof Banerjee andMajhi [19] near the horizon Finally it will benice to determine quantum corrected entropy of the BH in aconvenient form

6 Summary of the Work

This work is an attempt to study quantum corrections toHawking radiation of massless particle from a dynamical BHas well as for massive particle from a static BH At firstradial null geodesic tunnelling approach has been used withPainleve-type choice of coordinate system to derive semiclas-sical Hawking temperature Then full quantum mechanicalcalculations have performed writing action in a power seriesof the Planck constant ℏ to evaluate the quantum correc-tions to the Hawking temperature Subsequently quantumcorrected surface gravity has been calculated and it is foundthat one loop back reaction effects in the spacetime can beobtained by suitable choice of the arbitrary functions andparameters Finally an expression for the quantum correctedentropy of the BH has been evaluated It is found that dueto the presence of the arbitrary functions in the expressionfor entropy the leading order quantum correction may notbe logarithmic in nature On the other hand in the caseof Hawking radiation of massive particle from static BH itis found that Hawking temperature near the horizon doesnot depend on the mass term as predicted by Srinivasanand Padmanabhan [10] and Banerjee et al [16ndash18] Forfuture work we will try to find a solution for the partialdifferential equation (18) in a more simple form so that morephysical interpretations can be done from the BHparameters

Also it will be interesting to calculate temperature of theBH for tunnelling nonzero mass particle with full quantumcorrection and examine whether the result agrees with thatof Banerjee andMajhi [19] near the horizon Finally it will benice to determine quantum corrected entropy of the BH in aconvenient form

Conflict of Interests

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

Acknowledgments

The authors are thankful to IUCAA Pune India for theirwarm hospitality and research facilities as the work hasbeen done there during a visit Also Subenoy Chakrabortyacknowledges the UGC-DRS Programme in the Departmentof Mathematics Jadavpur University The author SubhajitSaha is thankful to UGC-BSR Programme of JadavpurUniversity for awarding research fellowship The authors arethankful to Ritabrata Biswas and Nairwita Mazumder fortheir help at the initial part of the work

References

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

[2] J B Hartle and S W Hawking ldquoPath-integral derivation ofblack-hole radiancerdquo Physical Review D vol 13 no 8 pp 2188ndash2203 1976

[3] J D Bekenstein ldquoGeneralized second law of thermodynamicsin black-hole physicsrdquo Physical Review D vol 9 p 2188 1974

[4] T Damour andR Ruffini ldquoBlack-hole evaporation in the Klein-Sauter-Heisenberg-Euler formalismrdquo Physical Review D vol 14p 332 1976

[5] S M Christensen and S A Fulling ldquoTrace anomalies and theHawking effectrdquo Physical ReviewD vol 15 no 8 pp 2088ndash21041977

[6] JW York ldquoBlack hole in thermal equilibriumwith a scalar fieldthe back-reactionrdquo Physical Review D vol 31 p 755 1985

[7] Z Zhao and J Y Zhu ldquoNernst theorem and Planck absoluteentropy of black holesrdquo Acta Physica Sinica vol 48 no 8 pp1558ndash1564 1999

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

[9] M K Parikh ldquoEnergy conservation and Hawking radiationrdquohttparxivorgabshep-th0402166

[10] K Srinivasan and T Padmanabhan ldquoParticle production andcomplex path analysisrdquo Physical Review D vol 60 no 2 ArticleID 024007 p 20 1999

[11] S Shankaranarayanan K Srinivasan and T PadmanabhanldquoMethod of complex paths and general covariance of HawkingradiationrdquoModern Physics Letters A Particles and Fields Grav-itation Cosmology Nuclear Physics vol 16 no 9 pp 571ndash5782001

[12] R Banerjee B R Majhi and S Samanta ldquoNoncommutativeblack hole thermodynamicsrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 77 no 12 Article ID124035 p 8 2008

Advances in High Energy Physics 9

[13] Q-Q Jiang S-Q Wu and X Cai ldquoHawking radiation as tun-neling from the Kerr and Kerr-Newman black holesrdquo PhysicalReview D vol 73 no 6 Article ID 064003 p 10 2006

[14] Y Hu J Zhang and Z Zhao ldquoMassive particlesrsquo Hawkingradiation via tunneling from the G H dilaton black holerdquoModern Physics Letters A Particles and Fields GravitationCosmology Nuclear Physics vol 21 no 28 pp 2143ndash2149 2006

[15] Z Xu and B Chen ldquoHawking radiation from general Kerr-(anti)de Sitter black holesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 75 no 2 Article ID 024041 p6 2007

[16] R Kerner and R B Mann ldquoFermions tunnelling from blackholesrdquo Classical and Quantum Gravity vol 25 no 9 Article ID095014 p 17 2008

[17] R D Kerner and R B Mann ldquoCharged fermions tunnellingfrom Kerr-Newman black holesrdquo Physics Letters B vol 665 no4 pp 277ndash283 2008

[18] D-Y Chen Q-Q Jiang and X-T Zu ldquoHawking radiation ofDirac particles via tunnelling from rotating black holes in deSitter spacesrdquo Physics Letters B vol 665 no 2-3 pp 106ndash1102008

[19] R Banerjee and B R Majhi ldquoQuantum tunneling beyondsemiclassical approximationrdquo Journal of High Energy Physicsno 6 Article ID 095 p 20 2008

[20] C Corda ldquoEffective temperature for black holesrdquo Journal ofHigh Energy Physics vol 1108 p 101 2011

[21] C Corda S H Hendi R Katebi and N O Schmidt ldquoEffectivestate Hawking radiation and quasi-normal modes for Kerrblack holesrdquo Journal of High Energy Physics no 6 p 008 2013

[22] H M Siahaan and T Triyanta ldquoSemiclassical methods forhawking radiation from a vaidya black holerdquo InternationalJournal of Modern Physics A vol 25 no 1 pp 145ndash153 2010

[23] B D Chowdhury ldquoProblems with tunneling of thin shells fromblack holesrdquo Pramana vol 70 no 1 pp 593ndash612 2008

[24] E T Akhmedov V Akhmedova and D Singleton ldquoHawkingtemperature in the tunneling picturerdquo Physics Letters B vol 642no 1-2 pp 124ndash128 2006

[25] E T Akhmedov T Pilling and D Singleton ldquoSubtleties in thequasi-classical calculation of Hawking radiationrdquo InternationalJournal of Modern Physics D Gravitation Astrophysics Cosmol-ogy vol 17 no 13-14 pp 2453ndash2458 2008

[26] R Banerjee and B R Majhi ldquoQuantum tunnelingand backreactionrdquo Physics Letters B vol 662 no 1 pp 62ndash65 2008

[27] C Corda ldquoEffective temperature Hawking radiation and quasi-normal modesrdquo International Journal of Modern Physics D vol21 no 11 Article ID 1242023 2012

[28] H-L Li and S-Z Yang ldquoHawking radiation from the chargedBTZ black hole with back-reactionrdquo Europhysics Letters vol 79no 2 Article ID 20001 p 3 2007

[29] C Corda S H Hendi R Katebi and N O Schmidt ldquoHawk-ing radiation-quasi-normal modes correspondence and effec-tive states for nonextremal Reissner-Nordstrom black holesrdquoAdvances in High Energy Physics vol 2014 Article ID 5278749 pages 2014

[30] Q-Q Jiang and S-Q Wu ldquoHawking radiation of chargedparticles as tunneling from Reissner-Nordstrom-de Sitter blackholes with a global monopolerdquo Physics Letters B vol 635 no2-3 pp 151ndash155 2006

[31] Y Hu J Zhang and Z Zhao ldquoMassive uncharged and chargedparticlesrsquo tunneling from the Horowitz-Strominger dilatonblack holerdquo International Journal of Modern Physics D Gravi-tation Astrophysics Cosmology vol 16 no 5 pp 847ndash855 2007

[32] C O Lousto and N G Sanchez ldquoBack reaction effects in blackhole spacetimes rdquo Physics Letters B vol 212 no 4 pp 411ndash4141988

[33] P Mitra ldquoHawking temperature from tunnelling formalismrdquoPhysics Letters B vol 648 no 2-3 pp 240ndash242 2007

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

Advances in High Energy Physics 5

Then from the principle of ldquodetailed balancerdquo [10ndash12] (whichstates that transitions between any two states take place withequal frequency in either direction at equilibrium) we write

119875emm = expminus1205960 (119905)

119879ℎ

119875in = expminus1205960 (119905)

119879ℎ

(34)

So comparing (33) and (34) the temperature of the BH isgiven by

119879ℎ=ℏ

4[1 + Σ

119896ℏ119896120596119896(119905)

1205960(119905)]

minus1

[Imint

119903

0

119889119903

radic119860119861

]

minus1

(35)

where

119879ℎ=ℏ

4[Imint

119903

0

119889119903

radic119860119861

]

minus1

(36)

is the usual Hawking temperature of the BH Thus due toquantum corrections the temperature of the BH is modifiedfrom the Hawking temperature and both temperatures arefunctions of 119905 and 119903 Note that (36) is the standard expressionfor semiclassical Hawking temperature and it is valid fornonspherical metric also However for spherical metric onecan use the Taylor series expansions (7) near the horizonand obtain 119879

119867as given in (11) by performing the contour

integration The ambiguity of factor of two (as mentionedearlier) in the Hawking temperature does not arise here

Further one may note that solutions (27) or (28) are theunique solutions to (18) or (21) except for a premultiplicationfactor This arbitrary multiplicative factor does not appear inthe expression for Hawking temperature only the particleenergy (120596

0) or 120596

119896is rescaled As quantum correction term

contains1205961198961205960 so it does not involve the arbitrarymultiplica-

tive factor and hence it is uniqueTo have some interpretation about the arbitrary functions

120596119896(119905) appearing in the quantum correction terms we make

use of dimensional analysis As 1198780has the dimension ℏ so the

arbitrary function 120596119896(119905) has the dimension ℏminus119896 In standard

choice of units namely 119866 = 119888 = 119870119861= 1 ℏ sim 119872

2

119901and so

120596119896sim 119872minus2119896 where119872 is the mass of the BH

Similar to the Hawking temperature the surface gravityof the BH is modified due to quantum corrections If 120581

119888is

the semiclassical surface gravity corresponding to Hawkingtemperature that is 120581

119888= 2120587119879

119867 then the quantum corrected

surface gravity 120581 = 2120587119879119867is related to the semiclassical value

by the relation

120581 = 120581119888[1 + Σ

119896ℏ119896120596119896(119905)

1205960(119905)]

minus1

(37)

Moreover based on the dimensional analysis if we choosefor simplicity

120596119896 (119905) =

1198861198961205960 (119905)

1198722119896 ldquo119886rdquo is a dimensionless parameter

(38)

then expression (37) is simplified to

120581 = 1205810(1 minus

ℏ119886

1198722)

minus1

(39)

This is related to the one loop back reaction effects in thespacetime [6 33] with the parameter 119886 corresponding to traceanomaly Higher order loop corrections to the surface gravitycan be obtained similarly by suitable choice of the functions120596119896(119905) For static BHs Banerjee and Majhi [19] have studied

these corrections in detail Lastly it is worth mentioning thatidentical result for BH temperature may be obtained if we usethe Painleve coordinate system as in the previous section

4 Entropy Function and Quantum Correction

We will now examine how the semiclassical Bekenstein-Hawking area law namely 119878BH = (1198604ℏ) (119860 is the area of thehorizon) is modified due to quantum corrections describedin the previous section The first law of the BH mechanicswhich is essentially the energy conservation relation relatedthe change of BHmass (119872) to the change of its entropy (119878BH)electric charge (119876) and angular momentum (119869) as

119889119872 = 119879ℎ119889119878BH + Φ119889119876 + Ω119889119869 (40)

Here Ω is the angular velocity and Φ is the electrostaticpotential So for nonrotating uncharged BHs the entropy hasthe simple form

119878BH = int119889119872

119879ℎ

(41)

or using (35) for 119879ℎ we get

119878BH = int[1 + Σ119896ℏ119896120596119896 (119905)

1205960 (119905)

]119889119872

119879119867

(42)

For choice (38) corresponding to one loop back reactioneffects we have from (42) the quantum corrected BH entropyas

119878BH = int[1 +119886ℏ

119872+1198862ℏ2

1198722+ sdot sdot sdot ]

119889119872

119879119867

(43)

The first term is the usual semiclassical Bekenstein-Hawkingentropy and the subsequent terms are the quantum cor-rections of different order For static BHs Banerjee andMajhi [19] have shown the correction terms of which theleading one gives the standard logarithmic correction On theother hand for nonstatic BHs as the proportionality factorsare time-dependent and arbitrary (see (42)) so the leadingorder correction term may not be logarithmic For futurework we will attempt to determine physical interpretation ofthe arbitrary time-dependent proportionality factors so thatquantum corrections may be evaluated

5 Hamilton-Jacobi Method for MassiveParticles Quantum Corrections

The KG equation for a scalar field 120595 describing a scalarparticle of mass119898

0has the form [10]

(◻ +1198982

0

ℏ2)120595 = 0 (44)

6 Advances in High Energy Physics

where the box operator ◻ is evaluated in the background ofa general static BH metric of the form

1198891199042= minus119860 (119903) 119889119905

2+1198891199032

119861 (119903)+ 1199032119889Ω2

2 (45)

The explicit form of the KG equation for the metric (45) is

minus1

119860

1205972120595

1205971199052+ 119861

1205972120595

1205971199032+1

2119860

120597 (119860119861)

120597119903

120597120595

120597119903+2119861

119903

120597120595

120597119903

+1

1199032 sin 120579120597

120597120579(sin 120579

120597120595

120597120579)

+1

1199032sin21205791205972120595

1205971206012=1198982

0

ℏ2120595 (119905 119903 120579 120601)

(46)

Due to spherical symmetry we can decompose 120601 in the form

120595 (119905 119903 120579 120601) = Φ (119905 119903) 119884119898

119897(120579 120601) (47)

where 120601 satisfies [10]

1

119860

1205972120595

1205971199052minus 119861

1205972120595

1205971199032minus1

2119860

120597 (119860119861)

120597119903

120597120595

120597119903minus2119861

119903

120597120595

120597119903

+ 119897 (119897 + 1)

1199032+1198982

0

ℏ2Φ (119905 119903) = 0

(48)

If we substitute the standard ansatz for the semiclassical wavefunction namely

120601 (119905 119903) = exp minus 119894ℏ119878 (119903 119905) (49)

then the action 119878 will satisfy the following differential equa-tion

[1

119860(120597119878

120597119905)

2

minus 119861(120597119878

120597119903)

2

minus 1198642

0(119903)]

minusℏ

119894[1

119860

1205972119878

1205971199052minus 1198612 1205972119878

1205971199032minus

1

2119860

120597 (119860119861)

120597119903+2119861

119903120597119878

120597119903] = 0

(50)

where 11986420= 1198982

0+ (11987121199032) and 1198712 = 119897(119897 + 1)ℏ2 is the angular

momentum To incorporate quantum corrections over thesemiclassical action we expand the actions in powers ofPlanck constant ℏ as

119878 (119903 119905) = 1198780 (119903 119905) + Σ119896ℏ119896119878119896 (119903 119905) (51)

where 1198780is the semiclassical action and 119896 is a positive integer

Now substituting this ansatz for 119878 in the differential equation

(50) and equating different powers of ℏ on both sides weobtain the following set of partial differential equations

ℏ01

119860(120597119878

120597119905)

2

minus 119861(120597119878

120597119903)

2

minus 1198642

0(119903) = 0 (52)

ℏ12

119860

1205971198780

120597119905

1205971198781

120597119905minus 2119861

1205971198780

120597119903

1205971198781

120597119903

minus1

119894[1

119860

12059721198780

1205971199052minus 1198612 12059721198780

1205971199032

minus1

2119860

120597 (119860119861)

120597119903+2119861

1199031205971198780

120597119903] = 0

(53)

ℏ21

119860(1205971198781

120597119905)

2

+2

119860

1205971198780

120597119905

1205971198782

120597119905minus 119861(

1205971198781

120597119903)

2

minus 21198611205971198780

120597119903

1205971198782

120597119903

minus1

119894[1

119860

12059721198781

1205971199052minus 1198612 12059721198781

1205971199032

minus1

2119860

120597 (119860119861)

120597119903+2119861

1199031205971198781

120597119903] = 0

(54)

and so onTo solve the semiclassical action 119878

0 we start with the

standard separable choice [10]

1198780 (119903 119905) = 1205960119905 + 1198630 (119903) (55)

Substituting this choice in (52) we obtain

1198630= plusmnint

119903

0

radic1205962

0minus 1198601198642

0

119860119861119889119903 = plusmn119868

0(say) (56)

where + or minus sign corresponds to absorption or emission ofscalar particle Now substituting this choice for 119878

0in (53) we

have the differential equation for first order corrections 1198781as

1205971198781

120597119905∓ radic119860119861radic1 minus

1198601198642

0

1205962

0

1205971198781

120597119903

∓radic119860119861

119894

[[

[

minus1

119903

radic1 minus1198601198642

0

1205962

+

(120597119860120597119903) (1198642

01205962) minus (2119860119871

21205962

01199033)

4radic1 minus (1198601198642

01205962)

]]

]

= 0

(57)

As before 1198781can be written in separable form as

1198781= 1205961119905 + 1198631 (119903) (58)

Advances in High Energy Physics 7

where

1198631= int

119903

0

119889119903

radic119860119861radic1 minus (1198601198642

01205962

0)

times[[

[

plusmn 1205961minusradic119860119861

119894

times

minus1

119903

radic1 minus1198601198642

0

1205962

+

(120597119860120597119903) (1198642

01205962) minus (2119860119871

21205962

01199033)

4radic1 minus (1198601198642

01205962)

]]

]

= plusmn1198681minus 1198682

(59)

Now due to complicated form if we retain terms up to firstorder quantum corrections that is

119878 = 1198780+ ℏ1198781= (1205960+ ℏ1205961) 119905 + 119863

0+ ℏ1198631 (119903) (60)

then the wave function denoting absorption and emissionsolutions of the KG equation (48) using (49) are of the form

120601abs = exp minus 119894ℏ(1205960+ ℏ1205961119905 + 1198680+ ℏ1198681minus ℏ1198682)

120601emm = exp minus 119894ℏ(1205960+ ℏ1205961119905 minus 1198680+ ℏ1198681minus ℏ1198682)

(61)

It is to be noted that in course of tunnelling across the hori-zon the coordinate nature changes that is more preciselythe signs of themetric coefficients in the (119903 119905)-hyperplane arealteredThus we can interpret this as that the time coordinatehas an imaginary part in crossing the horizon and accordinglythe temporal part has contribution to the probabilities [1933] Thus absorption and emission probabilities are given by

119875abs =1003816100381610038161003816120601in

1003816100381610038161003816

2= exp 2

ℏ(Im 120596

0+ ℏ1205961119905)

+ Im 1198680+ ℏ1198681minus Im ℏ119868

2

(62)

119875emm =1003816100381610038161003816120601out

1003816100381610038161003816

2= exp minus 119894

ℏ(Im 120596

0+ ℏ1205961119905)

minus Im 1198680+ ℏ1198681minus Im ℏ119868

2

(63)

In the classical limit ℏ rarr 0 there is no reflection so allingoing particles should be absorbed and hence [33]

limℏrarr0

119875abs = 1 (64)

So from (62) we must have

Im 1205960119905 = Im 119868

0 Im (120596

1119905 minus 1198682) = Im 119868

1 (65)

and as a result 119875emm simplifies to

119875emm = exp[[

[

minus41205960

ℏ

times Im

int

119903

0

119889119903

radic119860119861

(radic1 minus1198601198642

0

1205962

0

+ℏ (12059611205960)

radic1 minus (1198601198642

01205962

0)

)

]]

]

(66)

Using the principle of ldquodetailed balancerdquo [10 11 20 21]namely

119875emm = expminus 119864119879ℎ

119875in = expminus 119864119879ℎ

(67)

the temperature of the BH is given by

119879ℎ=ℏ119864

41205960

[[

[

Im

int

119903

0

119889119903

radic119860119861

times (radic1 minus1198601198642

0

1205962

0

+ℏ(12059611205960)

radic1 minus (1198601198642

01205962

0)

)

]]

]

minus1

(68)

where the semiclassical Hawking temperature of the BH hasthe expression

119879119867=ℏ119864

41205960

[

[

Imint

119903

0

119889119903

radic119860119861

radic1 minus1198601198642

0

1205962

0

]

]

minus1

(69)

Now to obtain themodified form of the surface gravity of theBH we start with the usual relation between surface gravityand Hawking temperature namely

120581119867= 2120587119879

119867 (70)

where 119879119867is given by (69)

So the quantum corrected surface gravity is given by

120581QC = 2120587119879ℎ (71)

Further for the present nonrotating uncharged static BHsusing the law of BH thermodynamics 119889119872 = 119879

ℎ119889119878 we have

the expression for the entropy of the BH as

119878BH = int41205960

ℏ119864(1 +

ℏ1205961

1205960

)119889119872int

119903

0

119889119903

radic119860119861

(72)

Finally it is easy to see from (68) that near the horizonthe presence of 1198642

0term can be neglected as it is multiplied

8 Advances in High Energy Physics

by themetric coefficient119860Therefore the quantum corrected(up to first order) temperature of the BH (in (68)) reduces to

119879ℎ=ℏ119864

41205960

(1 +ℏ1205961

1205960

)

minus1

[int

119903

0

119889119903

radic119860119861

]

minus1

(73)

and the Hawking temperature (given in (69)) becomes

119879119867=ℏ119864

41205960

[int

119903

0

119889119903

radic119860119861

]

minus1

(74)

So we have

119879ℎ= (1 +

ℏ1205961

1205960

)

minus1

119879119867 (75)

We see that if the energy of the tunnelling particle is chosen as1205960(ie 119864 = 120596

0) and 120596

1= 1205731119872 (for notations see Banerjee

and Majhi [19]) then the Hawking temperature given by(74) is the usual one derived for massless particles andthe quantum corrected temperature 119879

ℎgiven in (75) agrees

with that of Banerjee and Majhi [19] for massless particleTherefore Hawking temperature near the horizon remainsthe same for both massless and nonzero mass tunnellingparticles and it agrees with the claim of Srinivasan andPadmanabhan [10] and Banerjee and Majhi [19] For futurework it will be interesting to calculate the temperature of theBH for tunnelling nonzero mass particle with full quantumcorrections and examine whether the result agrees with thatof Banerjee andMajhi [19] near the horizon Finally it will benice to determine quantum corrected entropy of the BH in aconvenient form

6 Summary of the Work

This work is an attempt to study quantum corrections toHawking radiation of massless particle from a dynamical BHas well as for massive particle from a static BH At firstradial null geodesic tunnelling approach has been used withPainleve-type choice of coordinate system to derive semiclas-sical Hawking temperature Then full quantum mechanicalcalculations have performed writing action in a power seriesof the Planck constant ℏ to evaluate the quantum correc-tions to the Hawking temperature Subsequently quantumcorrected surface gravity has been calculated and it is foundthat one loop back reaction effects in the spacetime can beobtained by suitable choice of the arbitrary functions andparameters Finally an expression for the quantum correctedentropy of the BH has been evaluated It is found that dueto the presence of the arbitrary functions in the expressionfor entropy the leading order quantum correction may notbe logarithmic in nature On the other hand in the caseof Hawking radiation of massive particle from static BH itis found that Hawking temperature near the horizon doesnot depend on the mass term as predicted by Srinivasanand Padmanabhan [10] and Banerjee et al [16ndash18] Forfuture work we will try to find a solution for the partialdifferential equation (18) in a more simple form so that morephysical interpretations can be done from the BHparameters

Also it will be interesting to calculate temperature of theBH for tunnelling nonzero mass particle with full quantumcorrection and examine whether the result agrees with thatof Banerjee andMajhi [19] near the horizon Finally it will benice to determine quantum corrected entropy of the BH in aconvenient form

Conflict of Interests

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

Acknowledgments

The authors are thankful to IUCAA Pune India for theirwarm hospitality and research facilities as the work hasbeen done there during a visit Also Subenoy Chakrabortyacknowledges the UGC-DRS Programme in the Departmentof Mathematics Jadavpur University The author SubhajitSaha is thankful to UGC-BSR Programme of JadavpurUniversity for awarding research fellowship The authors arethankful to Ritabrata Biswas and Nairwita Mazumder fortheir help at the initial part of the work

References

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

[2] J B Hartle and S W Hawking ldquoPath-integral derivation ofblack-hole radiancerdquo Physical Review D vol 13 no 8 pp 2188ndash2203 1976

[3] J D Bekenstein ldquoGeneralized second law of thermodynamicsin black-hole physicsrdquo Physical Review D vol 9 p 2188 1974

[4] T Damour andR Ruffini ldquoBlack-hole evaporation in the Klein-Sauter-Heisenberg-Euler formalismrdquo Physical Review D vol 14p 332 1976

[5] S M Christensen and S A Fulling ldquoTrace anomalies and theHawking effectrdquo Physical ReviewD vol 15 no 8 pp 2088ndash21041977

[6] JW York ldquoBlack hole in thermal equilibriumwith a scalar fieldthe back-reactionrdquo Physical Review D vol 31 p 755 1985

[7] Z Zhao and J Y Zhu ldquoNernst theorem and Planck absoluteentropy of black holesrdquo Acta Physica Sinica vol 48 no 8 pp1558ndash1564 1999

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

[9] M K Parikh ldquoEnergy conservation and Hawking radiationrdquohttparxivorgabshep-th0402166

[10] K Srinivasan and T Padmanabhan ldquoParticle production andcomplex path analysisrdquo Physical Review D vol 60 no 2 ArticleID 024007 p 20 1999

[11] S Shankaranarayanan K Srinivasan and T PadmanabhanldquoMethod of complex paths and general covariance of HawkingradiationrdquoModern Physics Letters A Particles and Fields Grav-itation Cosmology Nuclear Physics vol 16 no 9 pp 571ndash5782001

[12] R Banerjee B R Majhi and S Samanta ldquoNoncommutativeblack hole thermodynamicsrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 77 no 12 Article ID124035 p 8 2008

Advances in High Energy Physics 9

[13] Q-Q Jiang S-Q Wu and X Cai ldquoHawking radiation as tun-neling from the Kerr and Kerr-Newman black holesrdquo PhysicalReview D vol 73 no 6 Article ID 064003 p 10 2006

[14] Y Hu J Zhang and Z Zhao ldquoMassive particlesrsquo Hawkingradiation via tunneling from the G H dilaton black holerdquoModern Physics Letters A Particles and Fields GravitationCosmology Nuclear Physics vol 21 no 28 pp 2143ndash2149 2006

[15] Z Xu and B Chen ldquoHawking radiation from general Kerr-(anti)de Sitter black holesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 75 no 2 Article ID 024041 p6 2007

[16] R Kerner and R B Mann ldquoFermions tunnelling from blackholesrdquo Classical and Quantum Gravity vol 25 no 9 Article ID095014 p 17 2008

[17] R D Kerner and R B Mann ldquoCharged fermions tunnellingfrom Kerr-Newman black holesrdquo Physics Letters B vol 665 no4 pp 277ndash283 2008

[18] D-Y Chen Q-Q Jiang and X-T Zu ldquoHawking radiation ofDirac particles via tunnelling from rotating black holes in deSitter spacesrdquo Physics Letters B vol 665 no 2-3 pp 106ndash1102008

[19] R Banerjee and B R Majhi ldquoQuantum tunneling beyondsemiclassical approximationrdquo Journal of High Energy Physicsno 6 Article ID 095 p 20 2008

[20] C Corda ldquoEffective temperature for black holesrdquo Journal ofHigh Energy Physics vol 1108 p 101 2011

[21] C Corda S H Hendi R Katebi and N O Schmidt ldquoEffectivestate Hawking radiation and quasi-normal modes for Kerrblack holesrdquo Journal of High Energy Physics no 6 p 008 2013

[22] H M Siahaan and T Triyanta ldquoSemiclassical methods forhawking radiation from a vaidya black holerdquo InternationalJournal of Modern Physics A vol 25 no 1 pp 145ndash153 2010

[23] B D Chowdhury ldquoProblems with tunneling of thin shells fromblack holesrdquo Pramana vol 70 no 1 pp 593ndash612 2008

[24] E T Akhmedov V Akhmedova and D Singleton ldquoHawkingtemperature in the tunneling picturerdquo Physics Letters B vol 642no 1-2 pp 124ndash128 2006

[25] E T Akhmedov T Pilling and D Singleton ldquoSubtleties in thequasi-classical calculation of Hawking radiationrdquo InternationalJournal of Modern Physics D Gravitation Astrophysics Cosmol-ogy vol 17 no 13-14 pp 2453ndash2458 2008

[26] R Banerjee and B R Majhi ldquoQuantum tunnelingand backreactionrdquo Physics Letters B vol 662 no 1 pp 62ndash65 2008

[27] C Corda ldquoEffective temperature Hawking radiation and quasi-normal modesrdquo International Journal of Modern Physics D vol21 no 11 Article ID 1242023 2012

[28] H-L Li and S-Z Yang ldquoHawking radiation from the chargedBTZ black hole with back-reactionrdquo Europhysics Letters vol 79no 2 Article ID 20001 p 3 2007

[29] C Corda S H Hendi R Katebi and N O Schmidt ldquoHawk-ing radiation-quasi-normal modes correspondence and effec-tive states for nonextremal Reissner-Nordstrom black holesrdquoAdvances in High Energy Physics vol 2014 Article ID 5278749 pages 2014

[30] Q-Q Jiang and S-Q Wu ldquoHawking radiation of chargedparticles as tunneling from Reissner-Nordstrom-de Sitter blackholes with a global monopolerdquo Physics Letters B vol 635 no2-3 pp 151ndash155 2006

[31] Y Hu J Zhang and Z Zhao ldquoMassive uncharged and chargedparticlesrsquo tunneling from the Horowitz-Strominger dilatonblack holerdquo International Journal of Modern Physics D Gravi-tation Astrophysics Cosmology vol 16 no 5 pp 847ndash855 2007

[32] C O Lousto and N G Sanchez ldquoBack reaction effects in blackhole spacetimes rdquo Physics Letters B vol 212 no 4 pp 411ndash4141988

[33] P Mitra ldquoHawking temperature from tunnelling formalismrdquoPhysics Letters B vol 648 no 2-3 pp 240ndash242 2007

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

6 Advances in High Energy Physics

where the box operator ◻ is evaluated in the background ofa general static BH metric of the form

1198891199042= minus119860 (119903) 119889119905

2+1198891199032

119861 (119903)+ 1199032119889Ω2

2 (45)

The explicit form of the KG equation for the metric (45) is

minus1

119860

1205972120595

1205971199052+ 119861

1205972120595

1205971199032+1

2119860

120597 (119860119861)

120597119903

120597120595

120597119903+2119861

119903

120597120595

120597119903

+1

1199032 sin 120579120597

120597120579(sin 120579

120597120595

120597120579)

+1

1199032sin21205791205972120595

1205971206012=1198982

0

ℏ2120595 (119905 119903 120579 120601)

(46)

Due to spherical symmetry we can decompose 120601 in the form

120595 (119905 119903 120579 120601) = Φ (119905 119903) 119884119898

119897(120579 120601) (47)

where 120601 satisfies [10]

1

119860

1205972120595

1205971199052minus 119861

1205972120595

1205971199032minus1

2119860

120597 (119860119861)

120597119903

120597120595

120597119903minus2119861

119903

120597120595

120597119903

+ 119897 (119897 + 1)

1199032+1198982

0

ℏ2Φ (119905 119903) = 0

(48)

If we substitute the standard ansatz for the semiclassical wavefunction namely

120601 (119905 119903) = exp minus 119894ℏ119878 (119903 119905) (49)

then the action 119878 will satisfy the following differential equa-tion

[1

119860(120597119878

120597119905)

2

minus 119861(120597119878

120597119903)

2

minus 1198642

0(119903)]

minusℏ

119894[1

119860

1205972119878

1205971199052minus 1198612 1205972119878

1205971199032minus

1

2119860

120597 (119860119861)

120597119903+2119861

119903120597119878

120597119903] = 0

(50)

where 11986420= 1198982

0+ (11987121199032) and 1198712 = 119897(119897 + 1)ℏ2 is the angular

momentum To incorporate quantum corrections over thesemiclassical action we expand the actions in powers ofPlanck constant ℏ as

119878 (119903 119905) = 1198780 (119903 119905) + Σ119896ℏ119896119878119896 (119903 119905) (51)

where 1198780is the semiclassical action and 119896 is a positive integer

Now substituting this ansatz for 119878 in the differential equation

(50) and equating different powers of ℏ on both sides weobtain the following set of partial differential equations

ℏ01

119860(120597119878

120597119905)

2

minus 119861(120597119878

120597119903)

2

minus 1198642

0(119903) = 0 (52)

ℏ12

119860

1205971198780

120597119905

1205971198781

120597119905minus 2119861

1205971198780

120597119903

1205971198781

120597119903

minus1

119894[1

119860

12059721198780

1205971199052minus 1198612 12059721198780

1205971199032

minus1

2119860

120597 (119860119861)

120597119903+2119861

1199031205971198780

120597119903] = 0

(53)

ℏ21

119860(1205971198781

120597119905)

2

+2

119860

1205971198780

120597119905

1205971198782

120597119905minus 119861(

1205971198781

120597119903)

2

minus 21198611205971198780

120597119903

1205971198782

120597119903

minus1

119894[1

119860

12059721198781

1205971199052minus 1198612 12059721198781

1205971199032

minus1

2119860

120597 (119860119861)

120597119903+2119861

1199031205971198781

120597119903] = 0

(54)

and so onTo solve the semiclassical action 119878

0 we start with the

standard separable choice [10]

1198780 (119903 119905) = 1205960119905 + 1198630 (119903) (55)

Substituting this choice in (52) we obtain

1198630= plusmnint

119903

0

radic1205962

0minus 1198601198642

0

119860119861119889119903 = plusmn119868

0(say) (56)

where + or minus sign corresponds to absorption or emission ofscalar particle Now substituting this choice for 119878

0in (53) we

have the differential equation for first order corrections 1198781as

1205971198781

120597119905∓ radic119860119861radic1 minus

1198601198642

0

1205962

0

1205971198781

120597119903

∓radic119860119861

119894

[[

[

minus1

119903

radic1 minus1198601198642

0

1205962

+

(120597119860120597119903) (1198642

01205962) minus (2119860119871

21205962

01199033)

4radic1 minus (1198601198642

01205962)

]]

]

= 0

(57)

As before 1198781can be written in separable form as

1198781= 1205961119905 + 1198631 (119903) (58)

Advances in High Energy Physics 7

where

1198631= int

119903

0

119889119903

radic119860119861radic1 minus (1198601198642

01205962

0)

times[[

[

plusmn 1205961minusradic119860119861

119894

times

minus1

119903

radic1 minus1198601198642

0

1205962

+

(120597119860120597119903) (1198642

01205962) minus (2119860119871

21205962

01199033)

4radic1 minus (1198601198642

01205962)

]]

]

= plusmn1198681minus 1198682

(59)

Now due to complicated form if we retain terms up to firstorder quantum corrections that is

119878 = 1198780+ ℏ1198781= (1205960+ ℏ1205961) 119905 + 119863

0+ ℏ1198631 (119903) (60)

then the wave function denoting absorption and emissionsolutions of the KG equation (48) using (49) are of the form

120601abs = exp minus 119894ℏ(1205960+ ℏ1205961119905 + 1198680+ ℏ1198681minus ℏ1198682)

120601emm = exp minus 119894ℏ(1205960+ ℏ1205961119905 minus 1198680+ ℏ1198681minus ℏ1198682)

(61)

It is to be noted that in course of tunnelling across the hori-zon the coordinate nature changes that is more preciselythe signs of themetric coefficients in the (119903 119905)-hyperplane arealteredThus we can interpret this as that the time coordinatehas an imaginary part in crossing the horizon and accordinglythe temporal part has contribution to the probabilities [1933] Thus absorption and emission probabilities are given by

119875abs =1003816100381610038161003816120601in

1003816100381610038161003816

2= exp 2

ℏ(Im 120596

0+ ℏ1205961119905)

+ Im 1198680+ ℏ1198681minus Im ℏ119868

2

(62)

119875emm =1003816100381610038161003816120601out

1003816100381610038161003816

2= exp minus 119894

ℏ(Im 120596

0+ ℏ1205961119905)

minus Im 1198680+ ℏ1198681minus Im ℏ119868

2

(63)

In the classical limit ℏ rarr 0 there is no reflection so allingoing particles should be absorbed and hence [33]

limℏrarr0

119875abs = 1 (64)

So from (62) we must have

Im 1205960119905 = Im 119868

0 Im (120596

1119905 minus 1198682) = Im 119868

1 (65)

and as a result 119875emm simplifies to

119875emm = exp[[

[

minus41205960

ℏ

times Im

int

119903

0

119889119903

radic119860119861

(radic1 minus1198601198642

0

1205962

0

+ℏ (12059611205960)

radic1 minus (1198601198642

01205962

0)

)

]]

]

(66)

Using the principle of ldquodetailed balancerdquo [10 11 20 21]namely

119875emm = expminus 119864119879ℎ

119875in = expminus 119864119879ℎ

(67)

the temperature of the BH is given by

119879ℎ=ℏ119864

41205960

[[

[

Im

int

119903

0

119889119903

radic119860119861

times (radic1 minus1198601198642

0

1205962

0

+ℏ(12059611205960)

radic1 minus (1198601198642

01205962

0)

)

]]

]

minus1

(68)

where the semiclassical Hawking temperature of the BH hasthe expression

119879119867=ℏ119864

41205960

[

[

Imint

119903

0

119889119903

radic119860119861

radic1 minus1198601198642

0

1205962

0

]

]

minus1

(69)

Now to obtain themodified form of the surface gravity of theBH we start with the usual relation between surface gravityand Hawking temperature namely

120581119867= 2120587119879

119867 (70)

where 119879119867is given by (69)

So the quantum corrected surface gravity is given by

120581QC = 2120587119879ℎ (71)

Further for the present nonrotating uncharged static BHsusing the law of BH thermodynamics 119889119872 = 119879

ℎ119889119878 we have

the expression for the entropy of the BH as

119878BH = int41205960

ℏ119864(1 +

ℏ1205961

1205960

)119889119872int

119903

0

119889119903

radic119860119861

(72)

Finally it is easy to see from (68) that near the horizonthe presence of 1198642

0term can be neglected as it is multiplied

8 Advances in High Energy Physics

by themetric coefficient119860Therefore the quantum corrected(up to first order) temperature of the BH (in (68)) reduces to

119879ℎ=ℏ119864

41205960

(1 +ℏ1205961

1205960

)

minus1

[int

119903

0

119889119903

radic119860119861

]

minus1

(73)

and the Hawking temperature (given in (69)) becomes

119879119867=ℏ119864

41205960

[int

119903

0

119889119903

radic119860119861

]

minus1

(74)

So we have

119879ℎ= (1 +

ℏ1205961

1205960

)

minus1

119879119867 (75)

We see that if the energy of the tunnelling particle is chosen as1205960(ie 119864 = 120596

0) and 120596

1= 1205731119872 (for notations see Banerjee

and Majhi [19]) then the Hawking temperature given by(74) is the usual one derived for massless particles andthe quantum corrected temperature 119879

ℎgiven in (75) agrees

with that of Banerjee and Majhi [19] for massless particleTherefore Hawking temperature near the horizon remainsthe same for both massless and nonzero mass tunnellingparticles and it agrees with the claim of Srinivasan andPadmanabhan [10] and Banerjee and Majhi [19] For futurework it will be interesting to calculate the temperature of theBH for tunnelling nonzero mass particle with full quantumcorrections and examine whether the result agrees with thatof Banerjee andMajhi [19] near the horizon Finally it will benice to determine quantum corrected entropy of the BH in aconvenient form

6 Summary of the Work

This work is an attempt to study quantum corrections toHawking radiation of massless particle from a dynamical BHas well as for massive particle from a static BH At firstradial null geodesic tunnelling approach has been used withPainleve-type choice of coordinate system to derive semiclas-sical Hawking temperature Then full quantum mechanicalcalculations have performed writing action in a power seriesof the Planck constant ℏ to evaluate the quantum correc-tions to the Hawking temperature Subsequently quantumcorrected surface gravity has been calculated and it is foundthat one loop back reaction effects in the spacetime can beobtained by suitable choice of the arbitrary functions andparameters Finally an expression for the quantum correctedentropy of the BH has been evaluated It is found that dueto the presence of the arbitrary functions in the expressionfor entropy the leading order quantum correction may notbe logarithmic in nature On the other hand in the caseof Hawking radiation of massive particle from static BH itis found that Hawking temperature near the horizon doesnot depend on the mass term as predicted by Srinivasanand Padmanabhan [10] and Banerjee et al [16ndash18] Forfuture work we will try to find a solution for the partialdifferential equation (18) in a more simple form so that morephysical interpretations can be done from the BHparameters

Also it will be interesting to calculate temperature of theBH for tunnelling nonzero mass particle with full quantumcorrection and examine whether the result agrees with thatof Banerjee andMajhi [19] near the horizon Finally it will benice to determine quantum corrected entropy of the BH in aconvenient form

Conflict of Interests

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

Acknowledgments

The authors are thankful to IUCAA Pune India for theirwarm hospitality and research facilities as the work hasbeen done there during a visit Also Subenoy Chakrabortyacknowledges the UGC-DRS Programme in the Departmentof Mathematics Jadavpur University The author SubhajitSaha is thankful to UGC-BSR Programme of JadavpurUniversity for awarding research fellowship The authors arethankful to Ritabrata Biswas and Nairwita Mazumder fortheir help at the initial part of the work

References

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

[2] J B Hartle and S W Hawking ldquoPath-integral derivation ofblack-hole radiancerdquo Physical Review D vol 13 no 8 pp 2188ndash2203 1976

[3] J D Bekenstein ldquoGeneralized second law of thermodynamicsin black-hole physicsrdquo Physical Review D vol 9 p 2188 1974

[4] T Damour andR Ruffini ldquoBlack-hole evaporation in the Klein-Sauter-Heisenberg-Euler formalismrdquo Physical Review D vol 14p 332 1976

[5] S M Christensen and S A Fulling ldquoTrace anomalies and theHawking effectrdquo Physical ReviewD vol 15 no 8 pp 2088ndash21041977

[6] JW York ldquoBlack hole in thermal equilibriumwith a scalar fieldthe back-reactionrdquo Physical Review D vol 31 p 755 1985

[7] Z Zhao and J Y Zhu ldquoNernst theorem and Planck absoluteentropy of black holesrdquo Acta Physica Sinica vol 48 no 8 pp1558ndash1564 1999

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

[9] M K Parikh ldquoEnergy conservation and Hawking radiationrdquohttparxivorgabshep-th0402166

[10] K Srinivasan and T Padmanabhan ldquoParticle production andcomplex path analysisrdquo Physical Review D vol 60 no 2 ArticleID 024007 p 20 1999

[11] S Shankaranarayanan K Srinivasan and T PadmanabhanldquoMethod of complex paths and general covariance of HawkingradiationrdquoModern Physics Letters A Particles and Fields Grav-itation Cosmology Nuclear Physics vol 16 no 9 pp 571ndash5782001

[12] R Banerjee B R Majhi and S Samanta ldquoNoncommutativeblack hole thermodynamicsrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 77 no 12 Article ID124035 p 8 2008

Advances in High Energy Physics 9

[13] Q-Q Jiang S-Q Wu and X Cai ldquoHawking radiation as tun-neling from the Kerr and Kerr-Newman black holesrdquo PhysicalReview D vol 73 no 6 Article ID 064003 p 10 2006

[14] Y Hu J Zhang and Z Zhao ldquoMassive particlesrsquo Hawkingradiation via tunneling from the G H dilaton black holerdquoModern Physics Letters A Particles and Fields GravitationCosmology Nuclear Physics vol 21 no 28 pp 2143ndash2149 2006

[15] Z Xu and B Chen ldquoHawking radiation from general Kerr-(anti)de Sitter black holesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 75 no 2 Article ID 024041 p6 2007

[16] R Kerner and R B Mann ldquoFermions tunnelling from blackholesrdquo Classical and Quantum Gravity vol 25 no 9 Article ID095014 p 17 2008

[17] R D Kerner and R B Mann ldquoCharged fermions tunnellingfrom Kerr-Newman black holesrdquo Physics Letters B vol 665 no4 pp 277ndash283 2008

[18] D-Y Chen Q-Q Jiang and X-T Zu ldquoHawking radiation ofDirac particles via tunnelling from rotating black holes in deSitter spacesrdquo Physics Letters B vol 665 no 2-3 pp 106ndash1102008

[19] R Banerjee and B R Majhi ldquoQuantum tunneling beyondsemiclassical approximationrdquo Journal of High Energy Physicsno 6 Article ID 095 p 20 2008

[20] C Corda ldquoEffective temperature for black holesrdquo Journal ofHigh Energy Physics vol 1108 p 101 2011

[21] C Corda S H Hendi R Katebi and N O Schmidt ldquoEffectivestate Hawking radiation and quasi-normal modes for Kerrblack holesrdquo Journal of High Energy Physics no 6 p 008 2013

[22] H M Siahaan and T Triyanta ldquoSemiclassical methods forhawking radiation from a vaidya black holerdquo InternationalJournal of Modern Physics A vol 25 no 1 pp 145ndash153 2010

[23] B D Chowdhury ldquoProblems with tunneling of thin shells fromblack holesrdquo Pramana vol 70 no 1 pp 593ndash612 2008

[24] E T Akhmedov V Akhmedova and D Singleton ldquoHawkingtemperature in the tunneling picturerdquo Physics Letters B vol 642no 1-2 pp 124ndash128 2006

[25] E T Akhmedov T Pilling and D Singleton ldquoSubtleties in thequasi-classical calculation of Hawking radiationrdquo InternationalJournal of Modern Physics D Gravitation Astrophysics Cosmol-ogy vol 17 no 13-14 pp 2453ndash2458 2008

[26] R Banerjee and B R Majhi ldquoQuantum tunnelingand backreactionrdquo Physics Letters B vol 662 no 1 pp 62ndash65 2008

[27] C Corda ldquoEffective temperature Hawking radiation and quasi-normal modesrdquo International Journal of Modern Physics D vol21 no 11 Article ID 1242023 2012

[28] H-L Li and S-Z Yang ldquoHawking radiation from the chargedBTZ black hole with back-reactionrdquo Europhysics Letters vol 79no 2 Article ID 20001 p 3 2007

[29] C Corda S H Hendi R Katebi and N O Schmidt ldquoHawk-ing radiation-quasi-normal modes correspondence and effec-tive states for nonextremal Reissner-Nordstrom black holesrdquoAdvances in High Energy Physics vol 2014 Article ID 5278749 pages 2014

[30] Q-Q Jiang and S-Q Wu ldquoHawking radiation of chargedparticles as tunneling from Reissner-Nordstrom-de Sitter blackholes with a global monopolerdquo Physics Letters B vol 635 no2-3 pp 151ndash155 2006

[31] Y Hu J Zhang and Z Zhao ldquoMassive uncharged and chargedparticlesrsquo tunneling from the Horowitz-Strominger dilatonblack holerdquo International Journal of Modern Physics D Gravi-tation Astrophysics Cosmology vol 16 no 5 pp 847ndash855 2007

[32] C O Lousto and N G Sanchez ldquoBack reaction effects in blackhole spacetimes rdquo Physics Letters B vol 212 no 4 pp 411ndash4141988

[33] P Mitra ldquoHawking temperature from tunnelling formalismrdquoPhysics Letters B vol 648 no 2-3 pp 240ndash242 2007

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

Advances in High Energy Physics 7

where

1198631= int

119903

0

119889119903

radic119860119861radic1 minus (1198601198642

01205962

0)

times[[

[

plusmn 1205961minusradic119860119861

119894

times

minus1

119903

radic1 minus1198601198642

0

1205962

+

(120597119860120597119903) (1198642

01205962) minus (2119860119871

21205962

01199033)

4radic1 minus (1198601198642

01205962)

]]

]

= plusmn1198681minus 1198682

(59)

Now due to complicated form if we retain terms up to firstorder quantum corrections that is

119878 = 1198780+ ℏ1198781= (1205960+ ℏ1205961) 119905 + 119863

0+ ℏ1198631 (119903) (60)

then the wave function denoting absorption and emissionsolutions of the KG equation (48) using (49) are of the form

120601abs = exp minus 119894ℏ(1205960+ ℏ1205961119905 + 1198680+ ℏ1198681minus ℏ1198682)

120601emm = exp minus 119894ℏ(1205960+ ℏ1205961119905 minus 1198680+ ℏ1198681minus ℏ1198682)

(61)

It is to be noted that in course of tunnelling across the hori-zon the coordinate nature changes that is more preciselythe signs of themetric coefficients in the (119903 119905)-hyperplane arealteredThus we can interpret this as that the time coordinatehas an imaginary part in crossing the horizon and accordinglythe temporal part has contribution to the probabilities [1933] Thus absorption and emission probabilities are given by

119875abs =1003816100381610038161003816120601in

1003816100381610038161003816

2= exp 2

ℏ(Im 120596

0+ ℏ1205961119905)

+ Im 1198680+ ℏ1198681minus Im ℏ119868

2

(62)

119875emm =1003816100381610038161003816120601out

1003816100381610038161003816

2= exp minus 119894

ℏ(Im 120596

0+ ℏ1205961119905)

minus Im 1198680+ ℏ1198681minus Im ℏ119868

2

(63)

In the classical limit ℏ rarr 0 there is no reflection so allingoing particles should be absorbed and hence [33]

limℏrarr0

119875abs = 1 (64)

So from (62) we must have

Im 1205960119905 = Im 119868

0 Im (120596

1119905 minus 1198682) = Im 119868

1 (65)

and as a result 119875emm simplifies to

119875emm = exp[[

[

minus41205960

ℏ

times Im

int

119903

0

119889119903

radic119860119861

(radic1 minus1198601198642

0

1205962

0

+ℏ (12059611205960)

radic1 minus (1198601198642

01205962

0)

)

]]

]

(66)

Using the principle of ldquodetailed balancerdquo [10 11 20 21]namely

119875emm = expminus 119864119879ℎ

119875in = expminus 119864119879ℎ

(67)

the temperature of the BH is given by

119879ℎ=ℏ119864

41205960

[[

[

Im

int

119903

0

119889119903

radic119860119861

times (radic1 minus1198601198642

0

1205962

0

+ℏ(12059611205960)

radic1 minus (1198601198642

01205962

0)

)

]]

]

minus1

(68)

where the semiclassical Hawking temperature of the BH hasthe expression

119879119867=ℏ119864

41205960

[

[

Imint

119903

0

119889119903

radic119860119861

radic1 minus1198601198642

0

1205962

0

]

]

minus1

(69)

Now to obtain themodified form of the surface gravity of theBH we start with the usual relation between surface gravityand Hawking temperature namely

120581119867= 2120587119879

119867 (70)

where 119879119867is given by (69)

So the quantum corrected surface gravity is given by

120581QC = 2120587119879ℎ (71)

Further for the present nonrotating uncharged static BHsusing the law of BH thermodynamics 119889119872 = 119879

ℎ119889119878 we have

the expression for the entropy of the BH as

119878BH = int41205960

ℏ119864(1 +

ℏ1205961

1205960

)119889119872int

119903

0

119889119903

radic119860119861

(72)

Finally it is easy to see from (68) that near the horizonthe presence of 1198642

0term can be neglected as it is multiplied

8 Advances in High Energy Physics

by themetric coefficient119860Therefore the quantum corrected(up to first order) temperature of the BH (in (68)) reduces to

119879ℎ=ℏ119864

41205960

(1 +ℏ1205961

1205960

)

minus1

[int

119903

0

119889119903

radic119860119861

]

minus1

(73)

and the Hawking temperature (given in (69)) becomes

119879119867=ℏ119864

41205960

[int

119903

0

119889119903

radic119860119861

]

minus1

(74)

So we have

119879ℎ= (1 +

ℏ1205961

1205960

)

minus1

119879119867 (75)

We see that if the energy of the tunnelling particle is chosen as1205960(ie 119864 = 120596

0) and 120596

1= 1205731119872 (for notations see Banerjee

and Majhi [19]) then the Hawking temperature given by(74) is the usual one derived for massless particles andthe quantum corrected temperature 119879

ℎgiven in (75) agrees

with that of Banerjee and Majhi [19] for massless particleTherefore Hawking temperature near the horizon remainsthe same for both massless and nonzero mass tunnellingparticles and it agrees with the claim of Srinivasan andPadmanabhan [10] and Banerjee and Majhi [19] For futurework it will be interesting to calculate the temperature of theBH for tunnelling nonzero mass particle with full quantumcorrections and examine whether the result agrees with thatof Banerjee andMajhi [19] near the horizon Finally it will benice to determine quantum corrected entropy of the BH in aconvenient form

6 Summary of the Work

This work is an attempt to study quantum corrections toHawking radiation of massless particle from a dynamical BHas well as for massive particle from a static BH At firstradial null geodesic tunnelling approach has been used withPainleve-type choice of coordinate system to derive semiclas-sical Hawking temperature Then full quantum mechanicalcalculations have performed writing action in a power seriesof the Planck constant ℏ to evaluate the quantum correc-tions to the Hawking temperature Subsequently quantumcorrected surface gravity has been calculated and it is foundthat one loop back reaction effects in the spacetime can beobtained by suitable choice of the arbitrary functions andparameters Finally an expression for the quantum correctedentropy of the BH has been evaluated It is found that dueto the presence of the arbitrary functions in the expressionfor entropy the leading order quantum correction may notbe logarithmic in nature On the other hand in the caseof Hawking radiation of massive particle from static BH itis found that Hawking temperature near the horizon doesnot depend on the mass term as predicted by Srinivasanand Padmanabhan [10] and Banerjee et al [16ndash18] Forfuture work we will try to find a solution for the partialdifferential equation (18) in a more simple form so that morephysical interpretations can be done from the BHparameters

Also it will be interesting to calculate temperature of theBH for tunnelling nonzero mass particle with full quantumcorrection and examine whether the result agrees with thatof Banerjee andMajhi [19] near the horizon Finally it will benice to determine quantum corrected entropy of the BH in aconvenient form

Conflict of Interests

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

Acknowledgments

The authors are thankful to IUCAA Pune India for theirwarm hospitality and research facilities as the work hasbeen done there during a visit Also Subenoy Chakrabortyacknowledges the UGC-DRS Programme in the Departmentof Mathematics Jadavpur University The author SubhajitSaha is thankful to UGC-BSR Programme of JadavpurUniversity for awarding research fellowship The authors arethankful to Ritabrata Biswas and Nairwita Mazumder fortheir help at the initial part of the work

References

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

[2] J B Hartle and S W Hawking ldquoPath-integral derivation ofblack-hole radiancerdquo Physical Review D vol 13 no 8 pp 2188ndash2203 1976

[3] J D Bekenstein ldquoGeneralized second law of thermodynamicsin black-hole physicsrdquo Physical Review D vol 9 p 2188 1974

[4] T Damour andR Ruffini ldquoBlack-hole evaporation in the Klein-Sauter-Heisenberg-Euler formalismrdquo Physical Review D vol 14p 332 1976

[5] S M Christensen and S A Fulling ldquoTrace anomalies and theHawking effectrdquo Physical ReviewD vol 15 no 8 pp 2088ndash21041977

[6] JW York ldquoBlack hole in thermal equilibriumwith a scalar fieldthe back-reactionrdquo Physical Review D vol 31 p 755 1985

[7] Z Zhao and J Y Zhu ldquoNernst theorem and Planck absoluteentropy of black holesrdquo Acta Physica Sinica vol 48 no 8 pp1558ndash1564 1999

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

[9] M K Parikh ldquoEnergy conservation and Hawking radiationrdquohttparxivorgabshep-th0402166

[10] K Srinivasan and T Padmanabhan ldquoParticle production andcomplex path analysisrdquo Physical Review D vol 60 no 2 ArticleID 024007 p 20 1999

[11] S Shankaranarayanan K Srinivasan and T PadmanabhanldquoMethod of complex paths and general covariance of HawkingradiationrdquoModern Physics Letters A Particles and Fields Grav-itation Cosmology Nuclear Physics vol 16 no 9 pp 571ndash5782001

[12] R Banerjee B R Majhi and S Samanta ldquoNoncommutativeblack hole thermodynamicsrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 77 no 12 Article ID124035 p 8 2008

Advances in High Energy Physics 9

[13] Q-Q Jiang S-Q Wu and X Cai ldquoHawking radiation as tun-neling from the Kerr and Kerr-Newman black holesrdquo PhysicalReview D vol 73 no 6 Article ID 064003 p 10 2006

[14] Y Hu J Zhang and Z Zhao ldquoMassive particlesrsquo Hawkingradiation via tunneling from the G H dilaton black holerdquoModern Physics Letters A Particles and Fields GravitationCosmology Nuclear Physics vol 21 no 28 pp 2143ndash2149 2006

[15] Z Xu and B Chen ldquoHawking radiation from general Kerr-(anti)de Sitter black holesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 75 no 2 Article ID 024041 p6 2007

[16] R Kerner and R B Mann ldquoFermions tunnelling from blackholesrdquo Classical and Quantum Gravity vol 25 no 9 Article ID095014 p 17 2008

[17] R D Kerner and R B Mann ldquoCharged fermions tunnellingfrom Kerr-Newman black holesrdquo Physics Letters B vol 665 no4 pp 277ndash283 2008

[18] D-Y Chen Q-Q Jiang and X-T Zu ldquoHawking radiation ofDirac particles via tunnelling from rotating black holes in deSitter spacesrdquo Physics Letters B vol 665 no 2-3 pp 106ndash1102008

[19] R Banerjee and B R Majhi ldquoQuantum tunneling beyondsemiclassical approximationrdquo Journal of High Energy Physicsno 6 Article ID 095 p 20 2008

[20] C Corda ldquoEffective temperature for black holesrdquo Journal ofHigh Energy Physics vol 1108 p 101 2011

[21] C Corda S H Hendi R Katebi and N O Schmidt ldquoEffectivestate Hawking radiation and quasi-normal modes for Kerrblack holesrdquo Journal of High Energy Physics no 6 p 008 2013

[22] H M Siahaan and T Triyanta ldquoSemiclassical methods forhawking radiation from a vaidya black holerdquo InternationalJournal of Modern Physics A vol 25 no 1 pp 145ndash153 2010

[23] B D Chowdhury ldquoProblems with tunneling of thin shells fromblack holesrdquo Pramana vol 70 no 1 pp 593ndash612 2008

[24] E T Akhmedov V Akhmedova and D Singleton ldquoHawkingtemperature in the tunneling picturerdquo Physics Letters B vol 642no 1-2 pp 124ndash128 2006

[25] E T Akhmedov T Pilling and D Singleton ldquoSubtleties in thequasi-classical calculation of Hawking radiationrdquo InternationalJournal of Modern Physics D Gravitation Astrophysics Cosmol-ogy vol 17 no 13-14 pp 2453ndash2458 2008

[26] R Banerjee and B R Majhi ldquoQuantum tunnelingand backreactionrdquo Physics Letters B vol 662 no 1 pp 62ndash65 2008

[27] C Corda ldquoEffective temperature Hawking radiation and quasi-normal modesrdquo International Journal of Modern Physics D vol21 no 11 Article ID 1242023 2012

[28] H-L Li and S-Z Yang ldquoHawking radiation from the chargedBTZ black hole with back-reactionrdquo Europhysics Letters vol 79no 2 Article ID 20001 p 3 2007

[29] C Corda S H Hendi R Katebi and N O Schmidt ldquoHawk-ing radiation-quasi-normal modes correspondence and effec-tive states for nonextremal Reissner-Nordstrom black holesrdquoAdvances in High Energy Physics vol 2014 Article ID 5278749 pages 2014

[30] Q-Q Jiang and S-Q Wu ldquoHawking radiation of chargedparticles as tunneling from Reissner-Nordstrom-de Sitter blackholes with a global monopolerdquo Physics Letters B vol 635 no2-3 pp 151ndash155 2006

[31] Y Hu J Zhang and Z Zhao ldquoMassive uncharged and chargedparticlesrsquo tunneling from the Horowitz-Strominger dilatonblack holerdquo International Journal of Modern Physics D Gravi-tation Astrophysics Cosmology vol 16 no 5 pp 847ndash855 2007

[32] C O Lousto and N G Sanchez ldquoBack reaction effects in blackhole spacetimes rdquo Physics Letters B vol 212 no 4 pp 411ndash4141988

[33] P Mitra ldquoHawking temperature from tunnelling formalismrdquoPhysics Letters B vol 648 no 2-3 pp 240ndash242 2007

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

8 Advances in High Energy Physics

by themetric coefficient119860Therefore the quantum corrected(up to first order) temperature of the BH (in (68)) reduces to

119879ℎ=ℏ119864

41205960

(1 +ℏ1205961

1205960

)

minus1

[int

119903

0

119889119903

radic119860119861

]

minus1

(73)

and the Hawking temperature (given in (69)) becomes

119879119867=ℏ119864

41205960

[int

119903

0

119889119903

radic119860119861

]

minus1

(74)

So we have

119879ℎ= (1 +

ℏ1205961

1205960

)

minus1

119879119867 (75)

We see that if the energy of the tunnelling particle is chosen as1205960(ie 119864 = 120596

0) and 120596

1= 1205731119872 (for notations see Banerjee

and Majhi [19]) then the Hawking temperature given by(74) is the usual one derived for massless particles andthe quantum corrected temperature 119879

ℎgiven in (75) agrees

with that of Banerjee and Majhi [19] for massless particleTherefore Hawking temperature near the horizon remainsthe same for both massless and nonzero mass tunnellingparticles and it agrees with the claim of Srinivasan andPadmanabhan [10] and Banerjee and Majhi [19] For futurework it will be interesting to calculate the temperature of theBH for tunnelling nonzero mass particle with full quantumcorrections and examine whether the result agrees with thatof Banerjee andMajhi [19] near the horizon Finally it will benice to determine quantum corrected entropy of the BH in aconvenient form

6 Summary of the Work

This work is an attempt to study quantum corrections toHawking radiation of massless particle from a dynamical BHas well as for massive particle from a static BH At firstradial null geodesic tunnelling approach has been used withPainleve-type choice of coordinate system to derive semiclas-sical Hawking temperature Then full quantum mechanicalcalculations have performed writing action in a power seriesof the Planck constant ℏ to evaluate the quantum correc-tions to the Hawking temperature Subsequently quantumcorrected surface gravity has been calculated and it is foundthat one loop back reaction effects in the spacetime can beobtained by suitable choice of the arbitrary functions andparameters Finally an expression for the quantum correctedentropy of the BH has been evaluated It is found that dueto the presence of the arbitrary functions in the expressionfor entropy the leading order quantum correction may notbe logarithmic in nature On the other hand in the caseof Hawking radiation of massive particle from static BH itis found that Hawking temperature near the horizon doesnot depend on the mass term as predicted by Srinivasanand Padmanabhan [10] and Banerjee et al [16ndash18] Forfuture work we will try to find a solution for the partialdifferential equation (18) in a more simple form so that morephysical interpretations can be done from the BHparameters

Also it will be interesting to calculate temperature of theBH for tunnelling nonzero mass particle with full quantumcorrection and examine whether the result agrees with thatof Banerjee andMajhi [19] near the horizon Finally it will benice to determine quantum corrected entropy of the BH in aconvenient form

Conflict of Interests

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

Acknowledgments

The authors are thankful to IUCAA Pune India for theirwarm hospitality and research facilities as the work hasbeen done there during a visit Also Subenoy Chakrabortyacknowledges the UGC-DRS Programme in the Departmentof Mathematics Jadavpur University The author SubhajitSaha is thankful to UGC-BSR Programme of JadavpurUniversity for awarding research fellowship The authors arethankful to Ritabrata Biswas and Nairwita Mazumder fortheir help at the initial part of the work

References

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

[2] J B Hartle and S W Hawking ldquoPath-integral derivation ofblack-hole radiancerdquo Physical Review D vol 13 no 8 pp 2188ndash2203 1976

[3] J D Bekenstein ldquoGeneralized second law of thermodynamicsin black-hole physicsrdquo Physical Review D vol 9 p 2188 1974

[4] T Damour andR Ruffini ldquoBlack-hole evaporation in the Klein-Sauter-Heisenberg-Euler formalismrdquo Physical Review D vol 14p 332 1976

[5] S M Christensen and S A Fulling ldquoTrace anomalies and theHawking effectrdquo Physical ReviewD vol 15 no 8 pp 2088ndash21041977

[6] JW York ldquoBlack hole in thermal equilibriumwith a scalar fieldthe back-reactionrdquo Physical Review D vol 31 p 755 1985

[7] Z Zhao and J Y Zhu ldquoNernst theorem and Planck absoluteentropy of black holesrdquo Acta Physica Sinica vol 48 no 8 pp1558ndash1564 1999

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

[9] M K Parikh ldquoEnergy conservation and Hawking radiationrdquohttparxivorgabshep-th0402166

[10] K Srinivasan and T Padmanabhan ldquoParticle production andcomplex path analysisrdquo Physical Review D vol 60 no 2 ArticleID 024007 p 20 1999

[11] S Shankaranarayanan K Srinivasan and T PadmanabhanldquoMethod of complex paths and general covariance of HawkingradiationrdquoModern Physics Letters A Particles and Fields Grav-itation Cosmology Nuclear Physics vol 16 no 9 pp 571ndash5782001

[12] R Banerjee B R Majhi and S Samanta ldquoNoncommutativeblack hole thermodynamicsrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 77 no 12 Article ID124035 p 8 2008

Advances in High Energy Physics 9

[13] Q-Q Jiang S-Q Wu and X Cai ldquoHawking radiation as tun-neling from the Kerr and Kerr-Newman black holesrdquo PhysicalReview D vol 73 no 6 Article ID 064003 p 10 2006

[14] Y Hu J Zhang and Z Zhao ldquoMassive particlesrsquo Hawkingradiation via tunneling from the G H dilaton black holerdquoModern Physics Letters A Particles and Fields GravitationCosmology Nuclear Physics vol 21 no 28 pp 2143ndash2149 2006

[15] Z Xu and B Chen ldquoHawking radiation from general Kerr-(anti)de Sitter black holesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 75 no 2 Article ID 024041 p6 2007

[16] R Kerner and R B Mann ldquoFermions tunnelling from blackholesrdquo Classical and Quantum Gravity vol 25 no 9 Article ID095014 p 17 2008

[17] R D Kerner and R B Mann ldquoCharged fermions tunnellingfrom Kerr-Newman black holesrdquo Physics Letters B vol 665 no4 pp 277ndash283 2008

[18] D-Y Chen Q-Q Jiang and X-T Zu ldquoHawking radiation ofDirac particles via tunnelling from rotating black holes in deSitter spacesrdquo Physics Letters B vol 665 no 2-3 pp 106ndash1102008

[19] R Banerjee and B R Majhi ldquoQuantum tunneling beyondsemiclassical approximationrdquo Journal of High Energy Physicsno 6 Article ID 095 p 20 2008

[20] C Corda ldquoEffective temperature for black holesrdquo Journal ofHigh Energy Physics vol 1108 p 101 2011

[21] C Corda S H Hendi R Katebi and N O Schmidt ldquoEffectivestate Hawking radiation and quasi-normal modes for Kerrblack holesrdquo Journal of High Energy Physics no 6 p 008 2013

[22] H M Siahaan and T Triyanta ldquoSemiclassical methods forhawking radiation from a vaidya black holerdquo InternationalJournal of Modern Physics A vol 25 no 1 pp 145ndash153 2010

[23] B D Chowdhury ldquoProblems with tunneling of thin shells fromblack holesrdquo Pramana vol 70 no 1 pp 593ndash612 2008

[24] E T Akhmedov V Akhmedova and D Singleton ldquoHawkingtemperature in the tunneling picturerdquo Physics Letters B vol 642no 1-2 pp 124ndash128 2006

[25] E T Akhmedov T Pilling and D Singleton ldquoSubtleties in thequasi-classical calculation of Hawking radiationrdquo InternationalJournal of Modern Physics D Gravitation Astrophysics Cosmol-ogy vol 17 no 13-14 pp 2453ndash2458 2008

[26] R Banerjee and B R Majhi ldquoQuantum tunnelingand backreactionrdquo Physics Letters B vol 662 no 1 pp 62ndash65 2008

[27] C Corda ldquoEffective temperature Hawking radiation and quasi-normal modesrdquo International Journal of Modern Physics D vol21 no 11 Article ID 1242023 2012

[28] H-L Li and S-Z Yang ldquoHawking radiation from the chargedBTZ black hole with back-reactionrdquo Europhysics Letters vol 79no 2 Article ID 20001 p 3 2007

[29] C Corda S H Hendi R Katebi and N O Schmidt ldquoHawk-ing radiation-quasi-normal modes correspondence and effec-tive states for nonextremal Reissner-Nordstrom black holesrdquoAdvances in High Energy Physics vol 2014 Article ID 5278749 pages 2014

[30] Q-Q Jiang and S-Q Wu ldquoHawking radiation of chargedparticles as tunneling from Reissner-Nordstrom-de Sitter blackholes with a global monopolerdquo Physics Letters B vol 635 no2-3 pp 151ndash155 2006

[31] Y Hu J Zhang and Z Zhao ldquoMassive uncharged and chargedparticlesrsquo tunneling from the Horowitz-Strominger dilatonblack holerdquo International Journal of Modern Physics D Gravi-tation Astrophysics Cosmology vol 16 no 5 pp 847ndash855 2007

[32] C O Lousto and N G Sanchez ldquoBack reaction effects in blackhole spacetimes rdquo Physics Letters B vol 212 no 4 pp 411ndash4141988

[33] P Mitra ldquoHawking temperature from tunnelling formalismrdquoPhysics Letters B vol 648 no 2-3 pp 240ndash242 2007

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

Advances in High Energy Physics 9

[13] Q-Q Jiang S-Q Wu and X Cai ldquoHawking radiation as tun-neling from the Kerr and Kerr-Newman black holesrdquo PhysicalReview D vol 73 no 6 Article ID 064003 p 10 2006

[14] Y Hu J Zhang and Z Zhao ldquoMassive particlesrsquo Hawkingradiation via tunneling from the G H dilaton black holerdquoModern Physics Letters A Particles and Fields GravitationCosmology Nuclear Physics vol 21 no 28 pp 2143ndash2149 2006

[15] Z Xu and B Chen ldquoHawking radiation from general Kerr-(anti)de Sitter black holesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 75 no 2 Article ID 024041 p6 2007

[16] R Kerner and R B Mann ldquoFermions tunnelling from blackholesrdquo Classical and Quantum Gravity vol 25 no 9 Article ID095014 p 17 2008

[17] R D Kerner and R B Mann ldquoCharged fermions tunnellingfrom Kerr-Newman black holesrdquo Physics Letters B vol 665 no4 pp 277ndash283 2008

[18] D-Y Chen Q-Q Jiang and X-T Zu ldquoHawking radiation ofDirac particles via tunnelling from rotating black holes in deSitter spacesrdquo Physics Letters B vol 665 no 2-3 pp 106ndash1102008

[19] R Banerjee and B R Majhi ldquoQuantum tunneling beyondsemiclassical approximationrdquo Journal of High Energy Physicsno 6 Article ID 095 p 20 2008

[20] C Corda ldquoEffective temperature for black holesrdquo Journal ofHigh Energy Physics vol 1108 p 101 2011

[21] C Corda S H Hendi R Katebi and N O Schmidt ldquoEffectivestate Hawking radiation and quasi-normal modes for Kerrblack holesrdquo Journal of High Energy Physics no 6 p 008 2013

[22] H M Siahaan and T Triyanta ldquoSemiclassical methods forhawking radiation from a vaidya black holerdquo InternationalJournal of Modern Physics A vol 25 no 1 pp 145ndash153 2010

[23] B D Chowdhury ldquoProblems with tunneling of thin shells fromblack holesrdquo Pramana vol 70 no 1 pp 593ndash612 2008

[24] E T Akhmedov V Akhmedova and D Singleton ldquoHawkingtemperature in the tunneling picturerdquo Physics Letters B vol 642no 1-2 pp 124ndash128 2006

[25] E T Akhmedov T Pilling and D Singleton ldquoSubtleties in thequasi-classical calculation of Hawking radiationrdquo InternationalJournal of Modern Physics D Gravitation Astrophysics Cosmol-ogy vol 17 no 13-14 pp 2453ndash2458 2008

[26] R Banerjee and B R Majhi ldquoQuantum tunnelingand backreactionrdquo Physics Letters B vol 662 no 1 pp 62ndash65 2008

[27] C Corda ldquoEffective temperature Hawking radiation and quasi-normal modesrdquo International Journal of Modern Physics D vol21 no 11 Article ID 1242023 2012

[28] H-L Li and S-Z Yang ldquoHawking radiation from the chargedBTZ black hole with back-reactionrdquo Europhysics Letters vol 79no 2 Article ID 20001 p 3 2007

[29] C Corda S H Hendi R Katebi and N O Schmidt ldquoHawk-ing radiation-quasi-normal modes correspondence and effec-tive states for nonextremal Reissner-Nordstrom black holesrdquoAdvances in High Energy Physics vol 2014 Article ID 5278749 pages 2014

[30] Q-Q Jiang and S-Q Wu ldquoHawking radiation of chargedparticles as tunneling from Reissner-Nordstrom-de Sitter blackholes with a global monopolerdquo Physics Letters B vol 635 no2-3 pp 151ndash155 2006

[31] Y Hu J Zhang and Z Zhao ldquoMassive uncharged and chargedparticlesrsquo tunneling from the Horowitz-Strominger dilatonblack holerdquo International Journal of Modern Physics D Gravi-tation Astrophysics Cosmology vol 16 no 5 pp 847ndash855 2007

[32] C O Lousto and N G Sanchez ldquoBack reaction effects in blackhole spacetimes rdquo Physics Letters B vol 212 no 4 pp 411ndash4141988

[33] P Mitra ldquoHawking temperature from tunnelling formalismrdquoPhysics Letters B vol 648 no 2-3 pp 240ndash242 2007

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

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