The tunneling spectrum of Einsein Born-Infeld Black Hole. W. Ren2

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1 Intenational Confeence on Engineeing Management Engineeing Education and Infomation Technology (EMEEIT 015) The tunneling spectum of Einsein Bon-Infeld Black Hole J Tang1 W Ren Y Han3 1 Aba teaches college Wenchuan 6300 SichuanChina Chinese People's Amed Police Foce Academy Chengdu SichuanChina 3 China West Nomal Univesity Nanchong Sichuan China KEYWORDS: enegy consevation; emission spectum; tunneling EBI black hole ABSTRACT: The tunneling emission spectum of massless paticles via tunneling fom the EinseinBon-Infeld black hole has been eseached in this pape It is shown that that when the paticle s self-gavitation inteaction is included the emission spectum of the EBI black hole is not pecisely themal but satisfies the unitay theoy INTRODUCTION In 1974 Hawking fist poved that the black hole can adiate paticle and the spectum is pecisely themal which had an impotant significance in the development of black hole physics[1] Since then seveal deivations of Hawking adiation appea in the liteatue but most of them ae based on quantum field theoy on a fixed backgound spacetime without consideing the fluctuation of the spacetime geomety When consideing enegy consevation it seems that the backgound geomety of a adiating black hole should be alteed with the loss of enegy but this dynamical effect is often neglected in fomal teatments Recently a new method to descibe Hawking adiation as a tunneling pocess whee a paticle moves in dynamical geomety was initiated by Kaus and Wilczek and developed by Paikh and Wilczek[3-5] Thee ae two significant points in the tunneling pictue Fistly they pointed out thee is no peexisting baie and the baie is ceated by the outgoing paticle itself Secondly enegy consevation which was often neglected in the fome teatments of Hawking adiation was included In the tunneling famewok they investigated Hawking adiation of static spheically symmetic Schwazschild black hole and Reissne- Nodstöm black hole The esult shows the exact adiation spectum deviates fom the pue themal but is consistent with an undelying unitay theoy Following this method many papes appeaed to suppot the Paikh- Wilczek s opinion and pesented a coect amendment to the exact emission spectum of black hole[6-15] In this pape we develop the Paikh-Wilczek s method to study the Hawking adiation of massless paticle via tunneling fom the Einsein-Bon-Infeld (EBI) black hole EBI BLACK HOLE IN DRAGGING COORDINATE SYSTEM The space-time metic fo the EBI black hole can be witten as ds = ( dt EBI asin θ dϕ ) ρ ρ d + ρ dθ sin θ + adt S ( + a ) dϕ (1) ρ whee t E BI is the coodinate time fo EBI spacetime and + = ( M + B ) + Q + a + C ρ = + a cos θ whee 015 The authos - Published by Atlantis Pess () 140

2 Intenational Confeence on Engineeing Management Engineeing Education and Infomation Technology (EMEEIT 015) J cq C = c ( ) M 0 in which c is a paamete When 0 the EBI black hole educes to Ke-Newman black hole The ADM mass is elated to the mass paamete of the black hole with B MADM = (1+ )M M Fom the null supe-suface equation we can obtain the event hoizon equation ie (3) ( M + B ) + Q + akn +C = 0 a= Solving Eq (3) we have B B h = M + ± M + Q a C (4) whee + and ae intenal and extenal hoizon of the black hole espectively The event hoizon aea of EBI black hole is given by Ah = da = gdθ dϕ = 4π ( h + a ) (5) whee ( g = sin θ h + a ) The infinite ed-shift suface is given by the equation g 00 = 0 ie a sin θ = 0 Obviously the infinite ed-shift suface is not consistent with the event hoizon so the geometical optical appoximation cannot apply hee In ode to make coincidence between the infinite ed-shift suface and the event hoizon we pefom dagging coodinate tansfomation as g dϕ ϕ& = = 03 = Ω (6) dt EBI g 33 fo the line element (1) yields ρ ˆ ds = g 00 dt EBI + d + ρ dθ (7) whee g03 ρ (8) g 00 = g 00 = g33 ( + a ) a sin θ Obviously in dagging coodinate system the infinite ed-shift suface coincides with the event hoizon GENERAL PAINLEVE COORDINATE AND QUANTUM TUNNELING In (7) thee is also a coodinate singulaity at the event hoizon so it is necessay to eliminate the singulaity Hee we pefom geneal Painleve coodinate tansfomation dtebi = dt + F ( θ ) d + G ( θ ) dθ (9) whee F ( θ ) and G ( θ ) ae two undeteminated function of and θ and satisfy condition of integability F ( θ ) G ( θ ) = (10) θ Substituting (9) into line element (7) and Consideing flat Euclidean space in adial the line element in geneal Painleve-EBI spacetime can be witten as 015 The authos - Published by Atlantis Pess 141

3 Intenational Confeence on Engineeing Management Engineeing Education and Infomation Technology (EMEEIT 015) ds = gˆ 00 dt + d ± gˆ 00 (1 g11 )dtd + g 00 G ( θ ) + g dθ + + g 00 (1 g11 )G ( θ ) ddθ + g 00 G ( θ ) + g dθ + + g 00 (1 g11 )G ( θ ) ddθ + g 00G ( θ ) dtdθ (11) whee + and denote the line element of outgoing and ingoing paticles at the event hoizon of the black hole espectively Since the tunneling pocesses take place nea the event hoizon we may conside a paticle tunneling fom the event hoizon as an ellipsoid shell and think the paticle should be still an ellipsoid shell duing the tunneling pocess So the line element can be witten as ds = gˆ 00 dt + d + gˆ 00 (1 g11 )dtd + g 00G ( θ ) dtdθ (1) Fom Eq(1) we can get the adial null geodesics in these coodinates & = g11 gˆ 00 (1 g11) (13) Accoding to quantum field theoy when a pai of vitual paticles spontaneously ceates inside just inside the hoizon the positive enegy vitual paticle can tunnel out the hoizon and mateializes as a eal paticle; altenatively fo a pai ceated just outside the hoizon the negative enegy paticles which ae fobidden outside the hoizon can tunnel inwads In eithe case the negative paticles ae absobed by black hole esulting in a decease of the mass of the black hole Consideing enegy consevation and angula momentum consevation when a massive paticle is tunneled out as an ellipsoid massive shell of enegy 1 + ω M (including the static enegy) and angula momentum the mass paametes will be change to B M 1+ ω M Thus afte the paticle tunnels out the event hoizon of the black hole becomes h' = 1 + ( M ω ) ± 1 + ( M ω ) Q a C (14) M M and the dagging angula velocity of the black hole is given by Ω h = a h + a (15) Since the event hoizon coincides with the infinite ed-shift suface geometical optics limit can be applied hee Accoding to the WKB appoximation the tunneling ate is ImS (16) whee the action Γ ~e 015 The authos - Published by Atlantis Pess 14

4 Intenational Confeence on Engineeing Management Engineeing Education and Infomation Technology (EMEEIT 015) f P ϕ f P dpϕ dϕ ImS = Im dpd (17) i 0 ϕi 0 whee i and f coespond to the event hoizons befoe and afte the paticle tunneling out Hee it is teated as two tuning points of potential baie The distance between them is detemined by enegy of the tunneling paticle Accoding to Hamilton equation we have dh & = dp ( ;ϕ Pϕ ) dh (18) dpϕ (ϕ ; P ) Substituting (18) into (17) the imaginay pat of the action is given by H f f dh Ω h dj Im S = Im d & & H i i ϕ& = ω = Im 0 f 1 + M (1 aω ) h i d ( M ω ) d & π = ( f i ) (19) Then the tunneling ate at the event hoizon is Γ ~ e ImS π ( ) =e = e S f i BH (0) Obviously when enegy consevation and angula momentum consevation ae consideed the exact spectum of massless paticle via tunneling fom EBI black hole is not pecisely themal but satisfies the unitay theoy CONCLUSIONS When self-gavitation inteaction is included the Hawking adiation of massless paticle via tunneling fom EBI black hole is not pecisely themal but satisfies the unitay theoy Next we will show in some special case Eq (0) can ecove the well-known esults (i) When 0 we have π M (M ω) +M M a Q (M ω) ( M ω) a Q Γ=e which is the tunneling ate of the Ke-Newman black hole (ii) When 0 Q=0 we have π M (M ω) +M M a (M ω) (M ω) a Γ= e which coesponds to the tunneling ate of the Ke black hole (iii) When 0 Q=0a=0 we have 4π M (M ω) Γ= e which is the tunneling ate of the Schwazschild black hole[3] REFERENCES [1] SW Hawking Natue (1974); Commun Math Phys (1975) 015 The authos - Published by Atlantis Pess 143

5 Intenational Confeence on Engineeing Management Engineeing Education and Infomation Technology (EMEEIT 015) [] SW Hawking Phys Rev D (005) [3] MK Paikh and F Wilczek Phys Rev Lett (000) [4] MK Paikh Int J Mod Phys D (004); Phys Lett B (00) [5] P Kaus and F Wilczek Nucl Phys B (1995) [6] S Hemming and E Keski-Vakkui Phys Rev D (001) [7] A J M Medved Phys Rev D (00) [8] J Y Zhang Y P Hu and Z Zhao hep-th/05111(005) [9] R G Cai Nucl Phys B68 375(00) [10] J Y Zhang and Z Zhao JHEP (005) [11] QQ Jiang Phys Rev D (008); Phys Lett B (008) [1] QQ Jiang Y Han X Cai JHEP (010) [13] QQ Jiang X Cai JHEP (010) [14] S Z Yang Chin Phys Lett 49(005) [15] S Z Yang Q Q Jiang and H L Li Chin Phys (005) 015 The authos - Published by Atlantis Pess 144