Ameican Jounal of Space Science 1 (): 94-98, 013 ISSN: 1948-997 013 Science Publicaions doi:10.3844/ajssp.013.94.98 Published Online 1 () 013 (hp://www.hescipub.com/ajss.oc) TUNNELING OF DIRAC PARTICLES FROM PHANTOM REISSNER NORDSTROM ADS BLACK HOLE Goha, H. Depamen of Mahemaics, Kaakoam Inenaional Univesiy Gilgi, Pakisan Received 013-10-19; Revised 013-11-09; Acceped 013-11-7 ABSTRACT We sudy he Hawking adiaion of Diac paicles fom Phanom Reissne Nodsom ADS black hole by unneling mehod. We use chage elaivisic Diac equaion o sudy he emission of such paicles. To solve Diac equaion we use WKB appoximaion and find he unneling pobabiliy of ougoing paicles. Finally we find he Hawking empeaue fo such ype of black holes. Keywods: ADS Black Holes, Quanum Tunneling, Hawking Radiaion, Diac Paicles Science Publicaions 1. INTRODUCITON Paicles and ani paicles ae ceaed pemanenly a he even hoizon of he black hole due o he high gaviaional fields and vacuum flucuaions. Sephan Hawking heoeically showed ha one of he paicle can go ou of he black hole by calculaing he Bogoliubov ansfomaion beween he iniial and final saes of ingoing and ougoing adiaion (Hawking, 1974; 1975; Biel and Davies, 198). In his way Wick oaion Mehod (Gibbons and Hawking, 1977a; 1977b), Anomaly mehod (Iso e al., 006) and mehods of dimensional educion (Umesu, 010) has been widely used by diffeen auhos o invesigae he Hawking adiaions fom diffeen black holes. Similaly Hawking adiaion as quanum unneling effec a even hoizons of he black holes is discussed by diffeen auhos and quanum unneling mehod (Paikh and Wilczek, 000; Padmanabhan, 004; Sinivasan and Padmanabhan, 1999; Shankaanaayanan e al., 00; Kaus and Wilczek, 009; 1995; Kene and Mann, 006; 007; 008; Rehman and Saifullah, 011; Gillani and Saifullah, 011; Ahmed and Saifullah, 011; Goha and Saifullah 01a, 01b, 013; Jan and Goha, 013) is obus mehod o discuss hese adiaion. We can apply his mehod o almos all ype of black holes in diffeen gaviy heoies fom highe dimensions o lowe dimensions (Ejaz e al., 013; Vagenas, 00; Medved and Vagenas, 005; Masuno and Umesu, 011; Kim, 94 011; Hod, 011; Wu and Peng, 011). In his sudy we have used he unneling mehod o discuss he emission of Diac paicles fom Phanom Reissne Nodsom ADS black hole and finally ge he Hawking empeaue fo his black hole.. PHANTOM REISSNER NORDSTROM ADS BLACK HOLE Phanom Reissne Nodsom ADS black hole (RN ADS black hole) is a soluion of Einsein field equaions in he pesence of cosmological consan. This soluion aises due o coupling of a field of spin 1 wih gaviaional field. The field of spin 1 may be he usual Maxwell one, o wih a conibuion of negaive enegy densiy, called phanom. The acion fo his heoy is given by (Jadim e al., 01) Equaion (1): 4 µ v µ v (1) S = d x g R η F F Λ whee, he fis pa of he acion is he Einsein Hilbe acion, second is he coupling wih phanom field of spin 1 fo η = -1 o coupling wih Maxwell field fo η = 1 and hid em is coupling wih cosmological consan, Λ. Fo Λ > 0, I behaves as De Sie (DS) and fo Λ < 0, I behaves as Ani De Sie (ADS). R is he Ricci scala and F µv = A µ,v --- A v,µ, whee A µ is he fou poenial. The line elemen fo such ype of black holes is given by Equaion ( and 3):
Goha, H. / Ameican Jounal of Space Science 1 (): 94-98, 013 1 ds g()d [g()] d (d sin d ) Whee: = θ θ φ () M Λ Q g() 1 3 = η (3) Hee, M and Q ae he mass and chage of he black hole. Fo η = 1, he black hole is called RN ADS black hole and when fo η = -1; i is called ani RN ADS black hole. The even hoizon can be found by puing g() = 0 so we have wo hoizon, and -. Fo RN ADS; we have 0 < - < and fo ani RN ADS; we have - < 0 <. Fo ani RN ADS; we have he even hoizon given by (Jadim e al., 01) Equaion (4): 1 6 1M = x x Λ Λ x Whee Equaion (5 o 8): (4) x = A B (5) Λ whee, m and q ae he mass and chage of he paicle. A µ is he elecomagneic fou poenial and ohe paamees ae defined as follow Equaion (11 and 1): i αβ Ω µ = Γµ αβ (11) i α β αβ = γ, γ 4 (1) Hee, γ µ maices saisfy [γ µ,γ v ] = g µv I, whee I is he ideniy maix. We use he following γ µ maices o sudy he unneling of Diac paicles fom he hoizons of ani RN ADS black hole Equaion (13 and 14): 1 i 0 0 σ g() 0 i σ 0 3 γ =, γ = g() 3 1 1 0 σ φ 1 0 σ γθ =, 1 γ = σ 0 sin θ σ 0 (13) (14) Pauli sigma, σ i, maices ae given by Equaion (15): 1 A 4 Λη Q = 3 y Λ (6) 0 1 0 i 1 0 1 0 i 0 0 1 1 3 σ =, σ =, σ = (15) B y 3 3 Λ = 3 (7) Fo a Diac paicle, he wave funcion ψ has wo spin saes namely spin up and spin down so we can ake he following ansaz fo his wave funcion ψ Equaion (17): y 36 M 4 Q = Λ ηλ (( 36Λ M 4ηΛQ ) 4( 1 4ηΛQ ) ) 1 3 The mass of he black hole is given by Equaion (9): (8) A(,, θ, φ) 0 i ψ = exp I (,, θ, φ) B(,, θ, φ) ħ 0 (16) Q Λ η 3 M = 1 3. QUANTUM TUNNELING (9) 0 C(,, θ, φ) i ψ = exp I (,, θ, φ) 0 ħ D(,, θ, φ) (17) To invesigae he Hawking adiaion of Diac paicles o spin half femions fom he black hole, we use he elaivisic chaged Diac equaion wih Diac field ψ = ψ (,, θ, φ) given by Equaion (10): ( µ µ µ ) iħ γ µ Ω iqa ψ mψ = 0 (10) Science Publicaions 95 Hee Ι is he acion fo classical ougoing ajecoy. Hee we deal wih only spin up case and fo spin down case he calculaions ae same wih signaue changes. Pu Equaion (16) in Equaion (10) and using lowes WKB appoximaion, we ge he following fou equaions in leading ode of ħ Equaion (18 o 1):
Goha, H. / Ameican Jounal of Space Science 1 (): 94-98, 013 ia( I qa ) g() B g() I ma 0 = ib( I qa ) A g() I mb = 0 g() B i θi I 0 φ = sin θ (18) (19) (0) W E qa ± = ± iπ (7) g( ) The oveall unneling pobabiliy is given by aking aio of ougoing and in-coming aes o ge he coec unneling ae is given by (Sinivasan and Padmanabhan, 1999; Shankaanaayanan e al., 00) Equaion (8 and 7): P Γ = P ou in,whee A i θi φi 0 = sin θ Science Publicaions (1) If we look a he symmeies of he space ime, and φ ae Killing fields in he especive diecion and we ae only dealing wih adial ajecoies, so we choose ansaz of he fom: I E W(, ) J = θ φ () E is he enegy of he paicle and J is he coesponding angula momenum of he paicle. Puing Equaion () in above equaions fo θ = θ o and consideing only he adial ajecoy so by assuming J = 0, we ge he following wo equaions: ia( E qa ) g() B g() w ma 0 = ib( E qa ) A g() w mb = 0 g() (3) (4) If we pu mass of he paicle, m = 0, we have wo soluions fo Equaion (3) and (4). Fo Equaion (6): A ib, we have W() W E qa g() = = = (5) A ib, we have W() W (E qa ) g() = = = (6) Plus and minus signs coespond o he ougoing and incoming paicles. Hee in Equaion (5), we have simple pole a = so we use eside heoy fo semi cicle and we ge: 96 P exp( Im(I)) = exp( Im(W )) (8) ou P exp( Im(I)) = exp( Im(W )) (9) in We can wie he oveall unneling pobabiliy as: Γ = exp( 4Im(W )) (30) By puing he value fom Equaion (7), we ge he unneling pobabiliy fo ougoing paicles as Equaion (31): E qa Γ = exp 4π g( ) O: qq π (E ) Γ = exp M Q Λ η 3 (31) (3) Fom Equaion (3) we can conclude ha he unneling pobabiliy does no depend upon he mass of he paicle. Compaing Equaion (3), wih he Bolzmann faco, we ge he Hawking empeaue fo his black hole and is given by: T bh 1 M Λ Q = η (33) π 3 3 Fo Massive paicle, m 0, we ge: A ie m g() = B ie m g()
Goha, H. / Ameican Jounal of Space Science 1 (): 94-98, 013 When we appoach nea he hoizon hen we A have = 1 and we ge he Hawking empeaue B simila o massless case. This is because a even hoizon massive paicle behaves like massless paicle fo moe deail (Kene and Mann, 008). We have also aken cae of he empoal conibuion o he imaginay pa of he acion so ha fo no value of he enegy, chage and angula momenum of he paicles will he unneling pobabiliies be geae han 1 (Akhmedov e al., 008; Akhmedova, 008) and hey will no violae uniaiy. If we look a hemodynamic elaions fo his black hole hen mass of he black hole is given by puing g() = 0 Equaion (34): Λ ηq M = 1 3 Science Publicaions (34) The enopy of he Black hole is given by Equaion (35): A S = = π (35) 4 Fo he elecic poenial V Equaion (36): M ηq s = V = Q Now we have: Q 3 (36) M 1 M Q Λ = T = S η (37) π 3 Which is fom he hemodynamic elaions and Equaion (37) is same as he Hawking empeaue calculaed fom quanum unneling mehod. Afe puing he value of M, We have he Hawking empeaue Equaion (38): 1 Q T 1 (38) π bh = Λ η 4 Which is consisen wih pevious lieaue (Jadim e al., 01). 97 4. CONCLUSION By ignoing he self gaviy and back eacion of he Diac paicle, we have discussed quanum unneling of Diac paicles fom he even hoizon of he phanom ADS black hole. Fo his pupose, we have used he elaivisic Diac Equaion and found he unneling pobabiliy of ougoing femions and we found he Hawking empeaue fo ADS black holes. Fom Equaion (3), we can say ha he unneling pobabiliy depends upon he chage of he paicle bu no on he mass of he paicle. We have jusified ou esuls by wiing he usual hemodynamic elaions and which ae same by using he unneling mehod. Las bu no leas, one can also find he unneling wih back eacion fom hese black holes, which will have vey ineesing esuls. 5. REFERENCES Ahmed, J. and K. Saifullah, 011. Hawking adiaion of Diac paicles fom black sings. J. Cosmol. Asopa. Phys., 08: 011-011. DOI: 10.1088/1475-7516/011/08/011 Akhmedov, E.T., T. Pilling and D. Singleon, 008. Subleies in he quasi-classical calculaion of hawking adiaion. In. J. Mod. Phys. D., 17: 453-458. DOI: 10.114/S01871808013947 Akhmedova, V., T. Pilling, A. de Gill and D. Singleon, 008. Tempoal conibuion o gaviaional WKBlike calculaions. Phys. Le. B, 666: 69-71. DOI: 10.1016/j.physleb.008.07.017 Biel, N.D. and P.C.W. Davies, 198. Quanum Fields in Cuved Space. 1s Edn., Cambidge Univesiy Pess, Cambidge, ISBN-10: 9780511663. DOI: 10.1017/CBO9780511663 Ejaz, A., H. Goha, H. Lin, K. Saifullah and S.T. Yau, 013. Quanum unneling fom hee-dimensional black holes. Phys. Le. B, 76: 87-833. DOI: 10.1016/j.physleb.013.09.015 Gibbons, G.W. and S.W. Hawking, 1977a. Cosmological even hoizons, he-modynamics and paicle ceaion. Phys. Rev., D, 15: 738-751. DOI: 10.1103/PhysRevD.15.738 Gibbons, G.W. and S.W. Hawking, 1977b. Acion inegals and paiion funcions in quanum gaviy. Phys. Rev. D, 15: 75-756. DOI: 10.1103/PhysRevD.15.75 Gillani, U.A. and K. Saifullah, 011. Tunneling of Diac paicles fom acceleaing and oaing black holes. Phys. Le. B, 699: 15-0. DOI: 10.1016/j.physleb.011.03.051
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