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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
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FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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
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Physics Research International
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Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
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Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
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PhotonicsJournal of
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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
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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
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Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
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AerodynamicsJournal of
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PhotonicsJournal of
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Journal of
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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
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FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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
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Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
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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
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FluidsJournal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
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Journal of
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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
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FluidsJournal of
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AstronomyAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
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GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Physics Research International
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Computational Methods in Physics
Journal of
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Soft MatterJournal of
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AerodynamicsJournal of
Volume 2014
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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