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Research ArticleModified Fermions Tunneling Radiation from Nonstationary,Axially Symmetric Kerr Black Hole

Jin Pu ,1,2 Kai Lin ,3 Xiao-Tao Zu ,1 and Shu-Zheng Yang 2

1School of Physics, University of Electronic Science and Technology of China, Chengdu 610054, Sichuan, China2College of Physics and Space Science, China West Normal University, Nanchong 637002, Sichuan, China3Hubei Subsurface Multi-Scale Imaging Key Laboratory, Institute of Geophysics and Geomatics, China University of Geosciences,Wuhan 430074, Hubei, China

Correspondence should be addressed to Shu-Zheng Yang; [email protected]

Received 4 June 2019; Accepted 14 July 2019; Published 22 July 2019

Guest Editor: Saibal Ray

Copyright © 2019 Jin Pu et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, by applying the deformeddispersion relation in quantumgravity theory,we study the correction of fermions’ tunnelingradiation fromnonstationary symmetric black holes. Firstly, themotion equation of fermions ismodified in the gravitational space-time. Based on the motion equation, the modified Hamilton-Jacobi equation has been obtained by a semiclassical approximationmethod.Then, the tunneling behavior of fermions at the event horizon of nonstationary symmetric Kerr black hole is investigated.Finally, the results show that, in the nonstationary symmetric background, the correction ofHawking temperature and the tunnelingrate are closely related to the angular parameters of the horizon of the black hole background.

1. Introduction

Since Hawking proposed that black holes can radiate ther-mally like a black body in 1974 [1, 2], a series of studieshave been carried out on static, stationary, and nonstationaryblack holes. Actually, Hawking thermal radiation is a purethermal radiation, and the radiation spectrum formed by thisradiation is a pure thermal radiation spectrum, which leadsto the problem of the information loss of black holes. In orderto explain the information loss paradox of black holes, Robin-son,Wilczek, Kraus, and Parikh havemodifiedHawking purethermal radiation spectrum and found that the informationwas conserved during Hawking tunneling radiation fromstatic and stationary black holes, by considering the self-gravitational interaction and the change of curved space-timebackground [3–8]. Subsequently, there are a series of studieson themassive particles and fermions via tunneling radiationfrom black holes [9–56]. However, the actual existence ofblack holes in the universe should be nonstationary, so theissues, such as the thermodynamic properties and the infor-mation conservation of nonstationary black holes, andmerg-ing process of black holes, deserve to be studied in depth.

On the other hand, quantum gravity theory suggests thatLorentz symmetrymay bemodified at high energy. Althoughthe dispersion relation theory at high energy has not yet beenfully established, it is generally accepted that the scale of thiscorrection term is equal to or close to the Planck scale. Wehave studied fermions’ quantum tunneling radiation fromstationary black holes by using the deformed dispersion rela-tion and obtained the very interesting results that there was acorrection in the tunneling radiation behavior [57]. However,the correction is only obtained for fermions’ tunneling radia-tion at the event horizon of stationary black holes. Therefore,we generalize the modified dispersion relation to study thequantum tunneling radiation of nonstationary symmetricKerr black holes in this paper and give an effective correctionof thermodynamic characteristics of the black holes.

The remainder of this paper is outlined as follows. InSection 2, by applying the modified dispersion relation inquantum gravity theory, we construct new Rarita-Schwingerequation and obtain the modified Hamilton-Jacobi equationof fermions by using semiclassical approximation method.In Section 3, the quantum tunneling radiation of fermionsfrom nonstationary symmetric Kerr black hole is modified

HindawiAdvances in High Energy PhysicsVolume 2019, Article ID 5864042, 7 pageshttps://doi.org/10.1155/2019/5864042

https://orcid.org/0000-0003-3553-8241https://orcid.org/0000-0001-9631-3586https://orcid.org/0000-0003-4050-8725https://orcid.org/0000-0003-2011-5430https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2019/5864042

2 Advances in High Energy Physics

correctly, and the tunneling rate and Hawking temperatureare modified. Section 4 ends up with some discussions andconclusions.

2. The Modified Hamilton-Jacobi Equation

Kerner and Mann studied the quantum tunneling radiationof the Dirac field using a semiclassical method [20, 21].Subsequently, this method was extended to study quan-tum tunneling radiation in various black holes. Since thekinematic equation of fermions is the Dirac equation butthe Dirac equation is related to the matrix equation, so anew method is proposed in the literature [22–25] to studythe tunneling radiation of Dirac particles in curved space-time of static and stationary black holes. This method isthat the Dirac equation is transformed into a simple matrixequation, and then this matrix equation is converted intothe Hamilton-Jacobi equation in the curved space-time byapplying the relationship between the gamma matrix andthe space-time metric. After this Hamilton-Jacobi equationwas proposed in 2009, it not only promoted the study ofquantum tunneling radiation of dynamic black holes, butalso effectively simplified the research work on quantumtunneling radiation of fermions. Recently, this Hamilton-Jacobi equation, combined with the modified Lorentz dis-persion relation, has been generalized to effectively revisethe quantum tunneling radiation of fermions from the eventhorizon of stationary axisymmetric Kerr-Newman de Sitterblack hole and has obtained very meaningful results [57].However, it only modifies the quantum tunneling radiationof fermions from the stationary black hole, while the realblack holes existing in the universe are nonstationary, sothe quantum tunneling radiation of fermions from the eventhorizon of the dynamic Kerr black hole is modified in thispaper by considering the correction of dispersion relation.

The Lorentz dispersion relation is considered to be oneof the basic relations in modern physics and related to thecorrelative theory research of general relativity and quantumfield theory. Research on the quantum gravity theory hasshown that the Lorentz relationship may be modified in thehigh energy field. In the study of string theory and quantumgravity theory, a dispersion relation has been proposed [58–66]: 𝑃20 = →𝑃2 + 𝑚2 − (𝐿𝑃0)𝛼 →𝑃2. (1)In the natural units, 𝑃0 and 𝑃 denote the energy andmomentum of particles, respectively. 𝑚 is the rest mass ofparticles and 𝐿 is a constant of the Planck scale. 𝛼 = 1 isused in the Liouville-string model [62–64]. Kruglov got themodified Dirac equation in the case of 𝛼 = 2 [67]. Themore general motion equation of fermions was proposedby Rarita and Schwinger and called the Rarita-Schwingerequation [68]. According to (1), we select 𝛼 = 2, so the Rarita-Schwinger equation in flat space-time is given by(𝛾𝜇𝜕𝜇 + 𝑚ℏ − 𝜎ℏ𝛾𝑡𝜕𝑡𝛾𝑗𝜕𝑗)𝜓𝛼1 ⋅⋅⋅𝛼𝑘 = 0, (2)where 𝛾𝜇 is the gamma matrix in flat space-time; 𝑗 and 𝜇denote space and time coordinates, respectively. According

to the relationship between the covariant derivative of thecurved space-time and the derivative of the flat space-time,the Rarita-Schwinger equation in the curved space-time ofnonstationary symmetric Kerr black hole can be expressed as(𝛾𝜇𝐷𝜇 + 𝑚ℏ − 𝜎ℏ𝛾v𝐷v𝛾𝑗𝐷𝑗)𝜓𝛼1⋅⋅⋅𝛼𝑘 = 0, (3)where 𝛾𝜇 is the gamma matrix in the curved space-time, andv denotes the advanced Eddington coordinate. In (3), when𝑘 = 0, we have 𝜙𝛼1∧𝛼𝑘 = 𝜙, in which case (3) represents theDirac equation of a spin of 1/2; when 𝑘 = 1, (3) describesthe motion equation of the fermions with the spin of 3/2.However, when 𝑚 = 0, the fermions with the spin of 3/2describe the Gravitino particle in the supersymmetry andsupergravity theory, which are a kind of fermions associatedwith the graviton, and the study on such particles is likely topromote the development of quantum gravity theory.

It is worth noting that (3) satisfies the following condi-tions: 𝛾𝜇𝛾] + 𝛾]𝛾𝜇 = 2𝑔𝜇]𝐼, (4)𝛾𝜇𝜓𝜇𝛼2⋅⋅⋅𝛼𝑘 = 𝐷𝜇𝜓𝜇𝛼2 ⋅⋅⋅𝛼𝑘 = 𝜓𝜇𝜇𝛼3⋅⋅⋅𝛼𝑘 , (5)where 𝐼 is the unit matrix. In (3),𝐷𝜇 is defined as𝐷𝜇 ≡ 𝜕𝜇 + Ω𝜇, (6)where Ω𝜇 is the spin connection. In (3), coupling constant𝜎 ≪ 1, and 𝜎ℏ𝛾v𝐷v𝛾𝑗𝐷𝑗 is very small quantity.

In order to study the tunneling radiation of fermionsin nonstationary curved space-time, 𝑆 is used to representthe action function of particles, and the wave function offermions is written as𝜓𝛼1 ⋅⋅⋅𝛼𝑘 = 𝜉𝛼1 ⋅⋅⋅𝛼𝑘𝑒(𝑖/ℏ)𝑆. (7)For nonstationary and axisymmetric curved space-time,there must be 𝜕𝜑𝑆 = 𝑛, (8)where 𝑛 is the angular momentum parameter of particles’tunneling radiation and a constant for nonstationary axisym-metric black holes. Substituting (6), (7), and (8) into (3), ℏis considered as a small quantity, and the lowest order isretained, so we obtain𝑖𝛾𝜇𝜕𝜇𝑆𝜉𝛼1 ⋅⋅⋅𝛼𝑘 + 𝑚𝜉𝛼1 ⋅⋅⋅𝛼𝑘 + 𝜎𝜕v𝑆𝛾v𝛾𝑗𝜕𝑗𝑆𝜉𝛼1 ⋅⋅⋅𝛼𝑘 = 0. (9)Because of 𝛾𝜇𝜕𝜇𝑆 = 𝛾v𝜕v𝑆 + 𝛾𝑗𝜕𝑗𝑆, (9) is abbreviated toΓ𝜇𝜕𝜇𝑆𝜉𝛼1 ⋅⋅⋅𝛼𝑘 +𝑀𝐷𝜉𝛼1 ⋅⋅⋅𝛼𝑘 = 0, (10)where Γ𝜇 = 𝛾𝜇 − 𝑖𝜎𝜕v𝑆𝛾v𝛾𝜇,𝑀𝐷 = 𝑚 − 𝜎 (𝜕v𝑆)2 𝑔vv. (11)We use premultiplication Γ]𝜕]𝑆 to (10) and getΓ]𝜕]𝑆Γ𝜇𝜕𝜇𝑆𝜉𝛼1 ⋅⋅⋅𝛼𝑘 +𝑀𝐷Γ]𝜕]𝑆𝜉𝛼1 ⋅⋅⋅𝛼𝑘 = 0, (12)where Γ]Γ𝜇 = 𝛾]𝛾𝜇 − 2𝑖𝜎𝜕𝛽𝑆𝛾𝛽𝛾]𝛾𝜇 + O (𝜎2) . (13)

Advances in High Energy Physics 3

Now, after exchanging 𝜇 and ] in (12) and comparing themwith (12), we get[𝛾𝜇𝛾] + 𝛾]𝛾𝜇2 𝜕𝜇𝑆𝜕]𝑆 + 𝑚2 − 2𝜎𝑚𝑔vv (𝜕v𝑆)2− 2𝑖𝜎𝜕v𝑆𝑔v𝛽𝜕𝛽𝑆𝛾𝜇𝜕𝜇𝑆] 𝜉𝛼1 ⋅⋅⋅𝛼𝑘 + O (𝜎2)= [𝑔𝜇]𝜕𝜇𝑆𝜕]𝑆 + 𝑚2 − 2𝜎𝑚𝑔vv (𝜕v𝑆)2− 2𝑖𝜎𝜕v𝑆𝑔v𝛽𝜕𝛽𝑆𝛾𝜇𝜕𝜇𝑆] 𝜉𝛼1 ⋅⋅⋅𝛼𝑘 + O (𝜎2) = 0.

(14)

Eq. (14) is further simplified to𝑖𝜎𝛾𝜇𝜕𝜇𝑆𝜉𝛼1 ⋅⋅⋅𝛼𝑘 +𝑀𝜉𝜉𝛼1⋅⋅⋅𝛼𝑘 = 0, (15)where 𝑀𝜉 = 𝑔𝜇]𝜕𝜇𝑆𝜕]𝑆 + 𝑚2 − 2𝜎𝑚𝑔vv (𝜕v𝑆)22𝜕v𝑆𝑔v𝛽𝜕𝛽𝑆 . (16)We use premultiplication −𝑖𝜎𝛾𝜇𝜕𝜇𝑆 for (15) and exchange 𝜇and ]. Then, we add it to (15) and divide that by 2, so we get[𝜎2𝑔𝜇]𝜕𝜇𝑆𝜕]𝑆 +𝑀2𝜉] 𝜉𝛼1⋅⋅⋅𝛼𝑘 = 0. (17)This is a matrix equation, actually an eigenmatrix equation.The condition that the equation has a nontrivial solution isthat the value of the determinant corresponding to its matrixis 0; that is, 𝜎2𝑔𝜇]𝜕𝜇𝑆𝜕]𝑆 +𝑀2𝜉 = 0. (18)Ignoring O(𝜎2), the modified Hamilton-Jacobi equation canbe obtained from the above equation as𝑔𝜇]𝜕𝜇𝑆𝜕]𝑆 + 𝑚2 − 2𝜎𝑚𝑔vv (𝜕v𝑆)2 = 0. (19)Obviously, modified Hamilton-Jacobi equation (19) isentirely different from the previously well-known Hamilton-Jacobi equation, with the addition of the modified term2𝜎𝑚𝑔vv(𝜕v𝑆)2. Equation (19), derived from the modifiedRarita-Schwinger equation, is not affected by the specificspin and can describe themotion equation of any fermions inthe semiclassical approximation method. For any fermionsin nonstationary curved space-time, it is convenient to studyand modify characteristics of quantum tunneling radiationof fermions as long as the properties of the curved space-timeand the action 𝑆 of fermions are known.3. Fermions’ Tunneling Radiation inNonstationary Symmetric Kerr Black Hole

In the advanced Eddington coordinate, the line element ofnonstationary symmetric Kerr black hole is expressed as𝑑𝑠2 = −(1 − 2𝑀𝑟𝜌2 )𝑑𝑟2 + 2𝑑v𝑑𝑟− 22𝑀𝑟𝑎 sin2𝜃𝜌2 𝑑v𝑑𝜑 − 2𝑎 sin2𝜃𝑑𝑟𝑑𝜑 + 𝜌2𝑑𝜃2

+ [(𝑟2 + 𝑎2) + 2𝑀𝑟𝑎2 sin2𝜃𝜌2 ] sin2𝜃𝑑𝜑2,(20)

where 𝜌2 = 𝑟2 + 𝑎2 cos2𝜃,𝑀 = 𝑀(v), 𝑎 = 𝑎(v). According to(20), the inverse tensors metric of the black hole is

𝑔𝜇] = (𝑔00 𝑔01 0 𝑔03𝑔10 𝑔11 0 𝑔130 0 𝑔22 0𝑔30 𝑔31 0 𝑔33), (21)where 𝑔00 = 𝑎2 sin2𝜃𝜌2 ,𝑔01 = 𝑔10 = 𝑟2 + 𝑎2𝜌2 ,𝑔03 = 𝑔30 = 𝑎𝜌2 ,𝑔13 = 𝑔31 = 𝑎𝜌2 ,𝑔11 = Δ𝜌2 ,𝑔22 = 1𝜌2 ,𝑔33 = 1𝜌2 sin2𝜃 ,Δ = 𝑟2 + 𝑞2 − 2𝑀𝑟.

(22)

According to (20), the null hypersurface equation of the blackhole is given by 𝑔𝜇] 𝜕𝑓𝜕𝑥𝜇 𝜕𝑓𝜕𝑥] = 0, (23)Substituting (22) into (23), the equation at the event horizonof the black hole is expressed as𝑎2𝑟2𝐻 sin2𝜃 + 𝑟2𝐻 − 2𝑚𝑟𝐻 + 𝑎2 + 𝑟2𝐻 − 2 (𝑟2𝐻 + 𝑎2) ̇𝑟𝐻= 0, (24)From (24), we have𝑟𝐻= 𝑀 + [𝑚2 − (1 − 2 ̇𝑟𝐻) (𝑎2 + 𝑎2 ̇𝑟𝐻 sin2𝜃 + 𝑟2𝐻 − 2𝑎2 ̇𝑟𝐻)]1/21 − 2 ̇𝑟𝐻 . (25)Obviously, the event horizon of the black hole 𝑟𝐻 is associatedwith 𝑎(v), 𝑀(v), 𝜕v𝑟|𝑟=𝑟𝐻 = ̇𝑟𝐻, and 𝜕𝜃𝑟|𝑟=𝑟𝐻 = 𝑟𝐻. Oncewe know the characteristics of the event horizon of the black


hole, we can study the quantum tunneling radiation at theevent horizon.

The motion equation of fermions is given by matrixequation (3). From the above research, we can conclude thatthe motion equation of any half-integer fermions can bereduced to (19), and (19) is the modified Hamilton-Jacobiequation, where 𝑆 is the main function of Hamilton, alsoknown as the action of fermions. Substituting (22) and (8)into (19), the motion equation of the half-integer fermions inspace-time of the black hole is obtained:𝑔00 (𝜕𝑆𝜕v)2 + 2𝜕𝑆𝜕v 𝜕𝑆𝜕𝑟 + 2𝑔03𝑛𝜕𝑆𝜕v + 2𝑔13𝑛𝜕𝑆𝜕𝑟+ 𝑔11 (𝜕𝑆𝜕𝑟)2 + 𝑔22 (𝜕𝑆𝜕𝜃)2 + 𝑔33𝑛2 + 𝑚2− 2𝜎𝑚𝑔00 (𝜕𝑆𝜕v)2 = 0.

(26)

Since the space-time of the black hole is axially symmetric,𝑛 is a constant according to 𝑛 = 𝜕𝑆/𝜕𝜑. Equation (26) isthe motion equation of fermions in nonstationary Kerr blackhole. Actually, (26) is amodifiedHamilton-Jacobi equation inthe nonstationary curved space-time, where 𝑆 = 𝑆(v, 𝑟, 𝜃). Inorder to solve the equation,we need to use the general tortoisecoordinate transformation as follows:𝑟⋆ = 𝑟 + 12𝜅 [𝑟 − 𝑟𝐻 (v0, 𝜃0)] ,

v⋆ = v − v0,𝜃⋆ = 𝜃 − 𝜃0. (27)According to (27), we have𝜕𝜕𝑟 = 2𝜅 (𝑟 − 𝑟𝐻) + 𝑟𝐻2𝜅 (𝑟 − 𝑟𝐻) 𝜕𝜕𝑟⋆ ,𝜕𝜕𝜃 = 𝜕𝜕𝜃⋆ − 𝑟𝐻𝑟𝐻2𝜅 (𝑟 − 𝑟𝐻) 𝜕𝜕𝑟⋆ ,𝜕𝜕v = 𝜕𝜕v⋆ − ̇𝑟𝐻𝑟𝐻2𝜅 (𝑟 − 𝑟𝐻) 𝜕𝜕𝑟⋆ .

(28)

Substituting (27) and (28) into (26) and noticing

𝑆 = 𝑆 (v⋆, 𝑟⋆, 𝜃⋆) ,𝜕𝑆𝜕v⋆ = −𝜔,𝜕𝑆𝜕𝜃⋆ = 𝑝𝜃,(29)

where 𝜔 denotes the energy of fermions’ tunneling radiation,𝑝𝜃 is 𝜃 component of the generalizedmomentumof fermions,𝜎 is a small quantity, and 𝜎 ̇𝑟2 is also a small quantity, theequation at the horizon of the black hole can be written as𝐴𝐷 ( 𝜕𝑆𝜕𝑟⋆)2 + 2 𝜕𝑆𝜕v⋆ 𝜕𝑆𝜕𝑟⋆ + 𝐵𝐷 𝜕𝜕𝑟⋆ + 2𝜅 (𝑟 − 𝑟𝐻) 𝐶𝐷= 0 (30)where𝐴 = 12𝜅 (𝑟 − 𝑟𝐻) {𝑔00 ̇𝑟𝐻 + 2 ̇𝑟𝐻 [2𝜅 (𝑟 − 𝑟𝐻) + 1]+ 𝑔11 [2𝜅 (𝑟 − 𝑟𝐻) + 1]2 + 𝑔11 [2𝜅 (𝑟 − 𝑟𝐻) + 1]2+ 𝑔22𝑟2𝐻} ,𝐵 = 𝑔22𝑝𝜃𝑟𝐻 − 𝑔13𝑛 − 𝑔03𝑛 ̇𝑟𝐻,𝐶 = 𝑔00𝜔2 + 𝑔22𝑝2𝜃 + 𝑔33𝑛 + 𝑚2 − 2𝜎𝑚𝑔00𝜔2,𝐷 = 𝑔00 ̇𝑟𝐻 − 𝑔01 + 2𝜎𝑚𝑔00 ̇𝑟𝐻.

(31)

When 𝑟 → 𝑟𝐻, we have𝐴( 𝜕𝑆𝜕𝑟⋆)2 + 2 (𝜔 − 𝜔0) 𝜕𝑆𝜕𝑟⋆ = 0, (32)

where

𝐴|𝑟→𝑟𝐻 = lim𝑟→𝑟𝐻v→v0𝜃→𝜃0

𝑔00 ̇𝑟2𝐻 − 2𝑔01 ̇𝑟𝐻 [2𝜅 (𝑟 − 𝑟𝐻) + 1] + 𝑔11 [2𝜅 (𝑟 − 𝑟𝐻) + 1]2 + 𝑔22𝑟2𝐻2𝜅 (𝑟 − 𝑟𝐻) (𝑔00 ̇𝑟𝐻 − 𝑔01 + 2𝜎𝑚𝑔00 ̇𝑟𝐻) = 1. (33)From 𝐴|𝑟→𝑟𝐻 = 1, we get𝜔0𝑟→𝑟𝐻 = 𝑔03𝑛 ̇𝑟𝐻 + 𝑔22𝑝𝜃 ̇𝑟𝐻 − 𝑔13𝑛𝑔00 ̇𝑟𝐻 − 𝑔01 + 2𝜎𝑚𝑔00 ̇𝑟𝐻= 𝑎𝑛 ̇𝑟𝐻 + 𝑝𝜃 ̇𝑟𝐻 − 𝑎𝑛𝑎2 sin2𝜃 ̇𝑟𝐻 − 𝑟2𝐻 − 𝑎2 + 2𝜎𝑚 ̇𝑟𝐻𝑎2 sin2𝜃

= 𝑎𝑛 ̇𝑟𝐻 + 𝑝𝜃 ̇𝑟𝐻 − 𝑎𝑛𝑎2 sin2𝜃 ̇𝑟𝐻 − 𝑟2𝐻 − 𝑎2 (1− 2𝜎𝑚 ̇𝑟𝐻𝑎2 sin2𝜃𝑎2 sin2𝜃 ̇𝑟𝐻 − 𝑟2𝐻 − 𝑎2 + O (𝜎2)) .(34)


So the event horizon surface gravity is given by

𝜅 = (1 − 2 ̇𝑟𝐻) 𝑟𝐻 −𝑀𝑎2 sin2𝜃0 ̇𝑟𝐻 (1 + 2𝜎𝑚) − (𝑟2𝐻 + 𝑎2) (1 − 2 ̇𝑟𝐻) + 4𝑀𝑟𝐻 − (𝑟2𝐻 + 𝑎2)= (1 − 2 ̇𝑟𝐻) 𝑟𝐻 −𝑀𝑎2 sin2𝜃0 ̇𝑟𝐻 (1 + 2𝜎𝑚) − (𝑟2𝐻 + 𝑎2) (1 − 2 ̇𝑟𝐻) + 4𝑀𝑟𝐻 (35)Wenotice 𝑟2𝐻(1−2 ̇𝑟𝐻)−2𝑀𝑟𝐻+𝑎2(1−2 ̇𝑟𝐻+ ̇𝑟2𝐻 sin2𝜃0)+𝑟2𝐻 = 0and get𝜅 = (1 − 2 ̇𝑟𝐻) 𝑟𝐻 −𝑀2𝑀𝑟𝐻 − (1 − ̇𝑟𝐻) ̇𝑟𝐻𝑎2 sin2𝜃0 + 𝑟2𝐻 + 2𝜎𝑚𝑎2 sin2𝜃0= (1 − 2 ̇𝑟𝐻) 𝑟𝐻 −𝑀(1 − 2 ̇𝑟𝐻) [𝑟2𝐻 + 𝑎2 (1 − ̇𝑟𝐻 sin2𝜃0)] + 𝑟2𝐻 + 2𝜎𝑚𝑎2 sin2𝜃0= (1 − 2 ̇𝑟𝐻) 𝑟𝐻 −𝑀(1 − 2 ̇𝑟𝐻) [𝑟2𝐻 + 𝑎2 (1 − ̇𝑟𝐻 sin2𝜃0)] + 𝑟2𝐻 × {1− 2𝜎𝑚𝑎2 sin2𝜃0(1 − 2 ̇𝑟𝐻) [𝑟2𝐻 + 𝑎2 (1 − ̇𝑟𝐻 sin2𝜃0)] + 𝑟2𝐻 + O (𝜎2)}

(36)

Obviously, the event horizon surface gravity is modified, andthemodified termdepends on 𝜃0. Itmeans that the correctionis made in different angle directions. Due to 𝜕𝑆/𝜕𝑟 = [1 +1/2𝜅(𝑟 − 𝑟𝐻)](𝜕𝑆/𝜕𝑟⋆), we have𝑆 = 𝑖𝜋2𝜅 [(𝜔 − 𝜔0) ± (𝜔 − 𝜔0)] . (37)Thus, the imaginary part of the total action and the quantumtunneling rate, respectively, are𝐼𝑚𝑆+ − 𝐼𝑚𝑆− = 𝑖𝜋2𝜅 [(𝜔 − 𝜔0) ± (𝜔 − 𝜔0)] , (38)Γ = Γ𝑒𝑚𝑖𝑠𝑠𝑖𝑜𝑛Γ𝑎𝑏𝑠𝑝𝑡𝑖𝑜𝑛 = exp [2𝜋𝜅 (𝜔 − 𝜔0)] (39)Here, as shown in (36), it is clear that 𝜅mentioned above is theevent horizon surface gravity of the black hole. So, the eventhorizon temperature of the black hole is given by𝑇|𝑟=𝑟𝐻 = 𝜅2𝜋 . (40)It is worth noting that the temperature (40) is the modifiedHawking temperature, since 𝜅 in (40) is the modified surfacegravity related to the correction term 2𝜎𝑚𝑎2sin2𝜃0. Obvi-ously, the correction of tunneling rate, surface gravity, andHawking temperature at the event horizon of the black holeare related not only to the rates of the event horizon changė𝑟𝐻, 𝑟𝐻, and𝑀(v) of the black hole, but also to the correctionof the angle parameter 𝜃0.4. Discussion

In this paper, we study the quantum tunneling radiation offermions in nonstationary curved space-time by combining

the modified Lorentz dispersion relation and obtain themodified character of quantum tunneling radiation related tothe effects of the Planck scale. The modified Dirac equationproposed by Kruglov is first extended to the modifiedRarita-Schwinger equation for the more general fermions,and the modified Hamilton-Jacobi equation of fermions isobtained in the semiclassical approximation method. Then,we study the quantum tunneling radiation of fermions incurved space-time of nonstationary symmetric Kerr blackhole using themodifiedHamilton-Jacobi equation and obtainthe correction of Hawking temperature and tunneling rate offermions. Interestingly, we found that the modified Hawkingtemperature at the event horizon of the black hole dependsnot only on the rates of the event horizon change ̇𝑟𝐻, 𝑟𝐻,and 𝑀(v) of the black hole, but also on the correctionof the angle parameter 𝜃0. It means that the correction ofHawking radiation not only is the radial property of the blackhole, but also is related to the angular property of the blackhole.

In the study of quantum tunneling radiation of blackholes, people first modified Hawking pure thermal radiationand then modified character of tunneling radiation fromstationary black holes. With the research on quantum gravityeffect, we combine the deformed dispersion relation tomodify the tunneling radiation of nonstationary symmetricKerr black hole effectively. We believe that the correction oftunneling radiation from other types of curved space-timewill yield some interesting results. This paper only provides amethod to modify quantum tunneling radiation, and furtherresearch is needed.

Data Availability

This paper is a theoretical research that does not involve dataprocessing. No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (GrantsNo. 11573022, No. 11805166) andby the starting funds of China West Normal University withGrants No. 17YC513 and No. 17C050.


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