An Improvement for Quantum Tunneling Radiation of Fermions in a Stationary Kerr-Newman Black Hole Spacetime
aa r X i v : . [ g r- q c ] F e b Prepared for submission to JHEP
An Improvement for Quantum Tunneling Radiationof Fermions in a Stationary Kerr-Newman BlackHole Spacetime
J. Zhang, a, M. Q. Liu, a Z. E. Liu a and S. Z. Yang b a College of Physics and Electronic Engineering, Qilu Normal University2 Wenbo Road, Jinan 250300, China b College of Physics and Space Science, China West Normal University1 Shida Road, Nanchong 637000 China
E-mail: [email protected]
Abstract:
By introducing a specific etheric-like vector in the Dirac equation with LorentzInvariance Violation (LIV) in the curved spacetime, an improved method for quantumtunneling radiation of fermions is proposed. As an example, we apply this new method to acharged axisymmetric Kerr-Newman black hole. Firstly, considering LIV theory, we derivea modified dynamical equation of fermion with spin 1/2 in the Kerr-Newman black holespacetime. Then we solve the equation and find the increase or decrease of black hole’sHawking temperature and entropy are related to constants a and c of the Dirac equationwith LIV in the curved spacetime. As c is positive, the new Hawking temperature is about √ a +2 cmk r √ a times higher than that without modification, but the entropy will decrease.We also make a brief discussion for the case of high spin fermions. Corresponding author. ontents
Black hole is a mysterious cosmic body with extremely intense gravity. With the detectionof LIGO and Virgo, more and more activities of black holes are found, so current theo-retical researches for black holes become more and more significant. Black holes can bedivided into three types: static black hole, stationary black hole and dynamic black hole.Hawking firstly proved that black holes have thermal radiation in theory by studying thequantum effect near the horizon of black holes [1, 2]. Hawking radiation effectively linksgravitational theory, quantum theory and thermodynamic statistics physics, and inspiredother researchers to study the thermodynamical evolution of black holes [3–6]. Parikh et al.pointed that the behavior of Hawking radiation can be regarded as a quantum tunneling[7, 8]. In their assumption, the event horizon of black hole is a potential barrier, virtual par-ticles yielded inside the horizon have a certain probability to escape from this barrier and beconverted to real particles radiating out from the black hole. Refs.[9–17] adopted the quan-tum tunneling method to investigate Hawking radiation for different types of black holes.Srinivasan et al. derived the Hamilton-Jacobi equation in curved spacetime from a scalarfield equation [11, 12]. Kerner and Mann et al. studied the tunneling radiation of Diracparticles using a semi-classical theory [18–20]. Lin and Yang proposed a new method tostudy the quantum tunneling radiation of fermions [21–24]. Their method can also be usedto study the quantum tunneling radiation of bosons. The results obtained in Refs.[21–24]show that the Hamilton-Jacobi equation in curved spacetime is a basic equation of particledynamic, which reflects the inherent consistency between Lorentz symmetry theory and theHamilton-Jacobi equation. Recently, we considered a light dispersion relationship derivedfrom string theory to research the modified quantum tunneling rates for spherical symmetryand axisymmetryic black holes[25, 26].General relativity is a gravitational theory that cannot be renormalized, so severalmodified gravitational theories have been proposed. Since the Lorentz Invariation Violation(LIV) may exist at high energy cases, various gravity models based on LIV have been– 1 –roposed [27–29]. In principle, LIV theory can solve the problem of renormalization ofgravitational theory. In addition, some studies on LIV suggest that the dark matter maybe just one of the effects of LIV theoretical models [30]. In string theory, electrodynamicsand non-abel theory, LIV has attracted extensive attention [31–33]. In the recent years,Dirac equation with LIV term in the flat spacetime has been studied by introducing etheric-like field terms [34, 35]. In this theory, Lorentz symmetry disappears due to the existence ofthe ether-like field. Therefore, some special properties which are inconsistent with Lorentzsymmetry theory will emerge at the high energy case. This is an interesting topic whichneeds to be explored in depth. On the other hand, the dynamics of fermions with LIV in thecurved spacetime is also an attractive subject worthy further study, which will influence thecorrection to quantum tunneling radiation of black hole. At present, the quantum tunnelingradiation of Dirac particles with etheric-like field terms has been investigated in sphericallysymmetric black holes [36]. We investigate the influence of different etheric-like vector u α on the solution of the modified Hamilton-Jacobi equation by using both the semi-classicalapproximation and beyond the semi-classical approximation[37].In this paper, the quantum tunneling radiation of fermions is modified in the axisym-metric charged Kerr-Newman black hole by considering a specific ether-like field vectorterm. Our paper is organized as follows: In Sec.2, considering LIV theory, the dynamicalequation of fermions with spin 1/2 is derived for Kerr-Newman black hole. In Sec.3, wesolve this dynamical equation and obtain the corrected physical quantities such as Hawkingtemperature and tunneling rate of the black hole. We make some discussions in Sec.4. In the Ref.[35], Nascimento et al. researched the particle’s action which includes LIV in theflat spacetime. Transferring the normal derivative to covariant derivative, and extendingthe commutation relation of gamma matrices ¯ γ µ and ¯ γ ν in the flat spacetime to that in thecurved sapcetime, we obtain Dirac equation of fermion, spin of which is 1/2, with LIV inthe curved spacetime as: { γ µ D µ [1 + ~ am ( γ µ D µ ) ] + b ~ γ + c ~ ( u α D α ) − m ~ ) } Ψ = 0 , (2.1)where m is the mass of fermion, a , b and c is small constants. Gamma matrices γ µ satisfythe condition: γ µ γ ν + γ ν γ µ = 2 g µν I, (2.2) γ γ µ + γ µ γ = 0 , (2.3)where g µν is the inverse metric tensor, and I is the unit matrix. In the flat spacetime,Eq.(2.2) reduces to ¯ γ µ ¯ γ ν + ¯ γ ν ¯ γ µ = 2 δ µν I , and Eq.(2.3) changes to ¯ γ ¯ γ µ + ¯ γ µ ¯ γ = 0 . InEq.(2.1) D µ = ∂ µ + i αβµ π αβ − i ~ qA µ , (2.4)– 2 –here q is the charge of fermion, and A µ is the electromagnetic potential of black hole. Thesecond term at the right side of Eq.(2.4) is spin connection which is a very small term in thedynamical equation and thus can be ignored. u α is an etheric-like vector, which satisfies: u α u α = const . (2.5)In order to solve the Dirac equation for fermions with spin 1/2, we assume its wave functionis Ψ = ψ AB e i ~ S = AB ! e i ~ S , (2.6)where A and B are matrix elements in the column matrix, S is the Hamilton principalfunction. Substituting Eq.(2.4) and Eq.(2.6) into Eq.(2.1), we get { iγ µ ( ∂ µ S − qA µ )[1 − am γ α γ β ( ∂ α S − qA α )( ∂ β S − qA β )] − cu α u β ( ∂ α S − qA α )( ∂ β S − qA β ) + bγ − m } Ψ = 0 . (2.7)Using Eq.(2.2), we have γ α γ β ( ∂ α S − qA α )( ∂ β S − qA β ) = g αβ ( ∂ α S − qA α )( ∂ β S − qA β ) . (2.8)Combining Eq.(2.8) and Eq.(2.7), they yields iγ µ ( ∂ µ S − qA µ )Ψ = [1 − am γ α γ β ( ∂ α S − qA α )( ∂ β S − qA β )] − { cu α u β ( ∂ α S − qA α )( ∂ β S − qA β ) + bγ + m } Ψ= [1 + am γ α γ β ( ∂ α S − qA α )( ∂ β S − qA β ) + O ( a )] { cu α u β ( ∂ α S − qA α )( ∂ β S − qA β ) + bγ + m } Ψ= [1 + ( cm u α u β + am g αβ )( ∂ α S − qA α )( ∂ β S − qA β ) − bm γ ] m Ψ . (2.9)Since b ≪ m , so bm γ is very small. Multiplying iγ ν ( ∂ ν S − qA ν ) at both sides of Eq.(2.9),we get − γ µ γ ν ( ∂ µ S − qA µ )( ∂ ν S − qA ν )Ψ = m + 2( cmu α u β + ag αβ )( ∂ α S − qA α )( ∂ β S − qA β )Ψ + O , (2.10)i. e., [ g µν ( ∂ µ S − qA µ )( ∂ ν S − qA ν )+2( cmu µ u ν + ag µν )( ∂ µ S − qA µ )( ∂ ν S − qA ν ) + m ]Ψ = 0 . (2.11)This is a matrix equation. The condition for this matrix equation to have nontrivial solu-tions is that the value of determinant of the wave function’s coefficient matrix is zero, i.e., ( g µν + 2 cmu µ u ν + 2 ag µν )( ∂ µ S − qA µ )( ∂ ν S − qA ν ) + m = 0 , (2.12)– 3 –r ( g µν + 2 cmu µ u ν a )( ∂ µ S − qA µ )( ∂ ν S − qA ν ) + m / (1 + 2 a ) = 0 . (2.13)Considering a is a very small constant and adopting Taylor expansion for the last term inthe Eq.(2.13), we get ( g µν + 2 cmu µ u ν a )( ∂ µ S − qA µ )( ∂ ν S − qA ν ) + m (1 − a ) = 0 . (2.14)From Eq.(2.1) to Eq.(2.14), we get a new dynamical equation for Dirac particles. Compar-ing with the normal Hamilton-Jacobi equation, one can find that, as cmu µ u ν = a = 0 ,Eq.(2.14) returns to the normal Hamilton-Jacobi equation. We call this deformed equa-tion as Dirac-Hamilton-Jacobi equation. The choice of the Hamilton principal function S depends on the selected coordinate and line element. In a stationary spacetime, it can begenerally expressed as S = S ( t, r, θ, ϕ ) . In the Boyer-Lindquist coordinate, the line element of Kerr-Newman black hole is writtenas [38] ds = ρ ( dr ∆ + dθ ) + sin θρ [( r + a kn ) dϕ − a kn dt ] − ∆ ρ [ dt − a kn sin θdϕ ] , (3.1)where ∆ = r + a kn − M r + Q ,ρ = r + a kn cos θ, (3.2)where M and a are the mass and angular momentum of unit mass of black hole. FromEq.(3.1) and Eq.(3.2), one can get the components of non-zero covariant metric tensors g tt = ρ ( a kn sin θ − ∆) ,g rr = ρ ∆ ,g θθ = ρ ,g ϕϕ = sin θρ [( r + a kn ) − ∆ a kn sin θ ] ,g tϕ = a kn sin θρ ( Q − M r ) , (3.3)and inverse metric tensors g tt = ρ ( a kn sin θ − r + a kn ∆ ) ,g rr = ∆ ρ ,g θθ = ρ ,g ϕϕ = ρ ( θ − a kn ∆ ) ,g tϕ = ρ ( Q − Mr ∆ ) . (3.4)– 4 –ccording to null super-surface equation g µν ∂f∂x µ ∂f∂x ν = 0 , (3.5)the event horizon of black hole r H satisfies following equation ∆ | r = r H = r H − M r H + a kn + Q = 0 . (3.6)The electromagnetic potential of the Kerr-Newman black hole A µ = ( A t , , , A ϕ ) ,A t = − Qrρ ,A ϕ = a kn Qr sin θρ . (3.7)Substituting Eq.(3.3) and (3.4) into Eq.(2.14), we get the dynamical equation of spin 1/2fermions with mass m and charge q in the curved spacetime, g tt ( ∂ t S − qA t ) + 2 g tϕ ( ∂ t S − qA t )( ∂ ϕ S − qA ϕ ) + g rr ( ∂ r S ) + g θθ ( ∂ θ S ) + g ϕϕ ( ∂ ϕ S − qA ϕ ) + (1 − a ) m + mc a [ u t u t ( ∂ t S − qA t ) + u r u r ( ∂ r S ) + u θ u θ ( ∂ θ S ) + u ϕ u ϕ ( ∂ ϕ S − qA ϕ ) + 2 u t u ϕ ( ∂ t S − qA t )( ∂ ϕ S − qA ϕ )+2 u t u r ( ∂ t S − qA t ) ∂ r S + 2 u t u θ ( ∂ t S − qA t ) ∂ θ S +2 u r u ϕ ∂ r S ( ∂ ϕ S − qA ϕ ) + 2 u θ u ϕ ∂ θ S ( ∂ ϕ S − qA ϕ )] = 0 . (3.8)After substituting the components of g µν into Eq.(3.8), multiplying ρ on both sides of theequation and merging the similar terms, the dynamical equation are as follows: ∆( ∂S∂r ) − [( r + a kn ) ∂S∂t + a kn ∂S∂ϕ + eQr ] + r m (1 − a ) + ( ∂S∂θ ) +( θ ∂S∂ϕ + a kn sin θ ∂S∂t ) + ρ m (1 − a ) cos θ + mcρ a [ u t u t ( ∂S∂t − qA t ) + u r u r ( ∂S∂r ) + u θ u θ ( ∂S∂θ ) + u ϕ u ϕ ( ∂S∂ ϕ − qA ϕ ) +2 u t u ϕ ( ∂S∂t − qA t )( ∂S∂ ϕ − qA ϕ ) + 2 u t u r ( ∂ t S − qA t ) ∂ r S +2 u t u θ ( ∂ t S − qA t ) ∂ θ S + 2 u r u ϕ ∂ r S ( ∂ ϕ S − qA ϕ )+2 u θ u ϕ ∂ θ S ( ∂ ϕ S − qA ϕ )] = 0 . (3.9)To solve the above equation, we must choose special u t , u r , u θ and u ϕ which must satisfyEq.(2.5). According to Eqs.(3.1)-(3.4) and the metric tensor of Kerr-Newman black hole,we choose the following u α : u t = k t ρ ( a kn sin θ − ∆) / ,u r = k r ∆ / ρ ,u θ = k θ ρ ,u ϕ = k ϕ ρ sin θa kn ( Q − Mr ) , (3.10)– 5 –here k t , k r , k θ , k ϕ are constants, which satisfies u α u α = k α , (3.11)where α denotes t, r, θ, ϕ . Substituting Eq.(3.10) and Eq.(3.6) into Eq.(3.9), and con-sidering the limit at the event horizon of black hole, we can get the dynamical equation ofspin / fermions at the event horizon of black hole, that is, ∆ | r → r H (1 + 2 cmk r a ) (cid:18) ∂S∂r (cid:19) | r → r H − (cid:20) ( r H + a kn ) ∂S∂t + a kn ∂S∂ϕ + qQr H (cid:21) = 0 . (3.12)Because the quantum tunneling radiation of a black hole is a property of the radial directionof the black hole, we care the radial component of the Hamilton principal function. FromEq.(3.12), ∂S∂r | r → r H = ± ( r H + a kn ) √ a ∆ | r → r H p a + 2 cmk r ∂S∂t + a kn ∂S∂ϕ + qQr H r H + a kn ! . (3.13)According to Eq.(3.1), we set S = − ωt + R ( r ) + Θ( θ ) + jϕ, (3.14)where ω is particle energy, j is a constant which describes the ϕ component of generalmomentum. Eq.(3.13) can be reducible to ∂S∂r | r → r H = dRdr | r → r H = ± ( r H + a kn ) √ a ∆ | r → r H p a + 2 cmk r ( ω − ω ) , (3.15)where ω = qQr H + a kn jr H + a kn , (3.16)where ω is the particle energy, ω is the chemical potential, which means the minimal energyof emission particles. Integrating the above equation from the inner side to the outer sideof r H with the residue theorem, we obtain S ± = R ± = ± R dr ( r H + a kn ) √ a ∆ | r → rH √ a +2 cmk r ( ω − ω ) , = ± iπ r H + a kn ) √ a ( r H − M ) √ a +2 cmk r ( ω − ω ) , (3.17)where subscript + and - denote outgoing and incoming module, respectively. According tothe tunneling theory of black hole, the tunneling rate is Γ = exp[ − S + − Im S − )]= exp( − ω − ω T ′ H ) , (3.18)where T ′ H is Hawking temperature after modification.– 6 – ′ H = ( r H − M )(1+2 a +2 cmk r ) / π ( r H + a kn ) √ a = T KNh (1 + 2 a + 2 cmk r ) / / √ a, (3.19)where T KNh is the Hawking temperature without LIV modification, T KNh = ( r H − M )2 π ( r H + a kn ) . (3.20) In this paper, we have modified the dynamical equation of Dirac particles in the curvedspacetime, considering LIV theory. Comparing Eq.(1) and Eq.(2.14), we simplify the com-plicate derivation process. By solving Eq.(2.14), the new Hamilton principal function S ofDirac particle is obtained. Since the quantum tunneling rate and Hawking temperature atthe horizon of black hole depend on the imaginary part of S , so the new quantities relatedto Hawking radiation are obtained naturally. These results are valuable for further study ofLIV theory and quantum gravitational theory. Because LIV theory modifies the Hawkingtemperature at the event horizon of Kerr-Newman black hole, it will lead to the correctionof black hole entropy. According to the first law of thermaldynamics of black hole, d M = T d S + V d J + U d Q, (4.1)where V and U are the rotation potential and electric potential of black hole respectively.From Eq.(3.19) and Eq.(3.20), the entropy with LIV theory correction in the Kerr-Newmanblack hole is S H = Z dS H = R dM − V dJ − UdQT H = R √ a √ a +2 cmk r dS KNh , (4.2)where S KNh denotes the entropy without modification, dS KNh = dM − V dJ − U dQT
KNh . (4.3)It can be seen from Eq.(3.19) and Eq.(4.2) that the increase or decrease of black hole’sHawking temperature and entropy are related to the value of constants a and c in Eq.(2.1).As c is positive, the new Hawking temperature is about √ a +2 cmk r √ a times higher than thatwithout modification, but the entropy will decrease. Above results are from semi-classicalapproximation and based on spin / fermions. Beyond semi-classical approximation canrefer our recent paper [37]. For spin / fermions, the wave function in Eq.(2.1) should besubstituted by Ψ = A V B V ! e i ~ S , (4.4)– 7 –here A V = [ A B ] T , B V = [ A B ] [24]. For arbitrary spin fermions, Ψ in Eq.(2.1) should besubstituted by Ψ α ...α k , where the value of α k corresponds to different spin. The larger α k ,the higher the spin is. Furthermore, the above conclusions are also applicable to the otherspherically symmetric and axisymmetric charged black holes. We will do further researchat this aspect in the future. Acknowledgments
This work is supported by the National Natural Science Foundation of China (grant U2031121),the Science Foundation of Sichuan Science and Technology Department(grant 2018JY0502)and the Natural Science Foundation of Shandong Province (grant ZR2020MA063, ZR2019MA059).
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