Tunneling Radiation of Massive Vector Bosons from Dilaton Black Holes

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1 Commun. Theor. Phys. 66 (06) Vol. 66, No., July, 06 Tunneling Radiation of Massive Vector Bosons from Dilaton Black Holes Ran Li (Ó ), Jun-Kun Zhao ( ±), and Xing-Hua Wu ( Ù) Department of Physics, Henan Normal University, Xinxiang , China (Received April 8, 06; revised manuscript received May 6, 06) Abstract It is well known that Hawking radiation can be treated as a quantum tunneling process of particles from the event horizon of black hole. In this paper, we attempt to apply the massive vector bosons tunneling method to study the Hawking radiation from the non-rotating and rotating dilaton black holes. Starting with the Proca field equation that govern the dynamics of massive vector bosons, we derive the tunneling probabilities and radiation spectrums of the emitted vector bosons from the static spherical symmetric dilatonic black hole, the rotating Kaluza Klein black hole, and the rotating Kerr Sen black hole. Comparing the results with the blackbody spectrum, we satisfactorily reproduce the Hawking temperatures of these dilaton black holes, which are consistent with the previous results in the literature. PACS numbers: Dy, Sq Key words: vector boson, tunneling, Hawking radiation, dilaton black hole Introduction The pioneering work ] by Hawking, that black hole can radiate particles characterized by the thermal spectrum with the temperature T = κ/π, where κ is the surface gravity of the black hole, has established the remarkable connection between general relativity, quantum field theory, and statistical thermodynamics. It is widely believed that a deeper understanding of Hawking radiation may shed some lights on seeking the underlying theory of quantum gravity. The original technique employed by Hawking, i.e. the method of quantum field theory on curved spacetime, is difficult to be generalized to study other more general complicated black hole backgrounds. Since then, several derivations of Hawking radiation have been proposed in the literature, including the Damour Ruffini method, ] trace anomaly method, 3] gravitational anomaly method, 45] and quantum tunneling method. 6] These methods can be used to reproduce the Hawking temperatures of black holes correctly. In recent years, the semi-classical quantum tunneling method, which treats Hawking radiation as a tunneling process 6] of particles from the event horizon, has already attracted a lot of attention. In this method, the imaginary part of the action of emitted particle is calculated by using the null geodesic equation. Zhang and Zhao have extended this method to Reissner Nordström black hole 7] and Kerr Newman black hole 8] by treating the emitted particle as de Broglie wave. Angheben et al. 9] also proposed an alternative method to derive the Hawking radiation by calculating the particles classical action from the Hamilton Jacobi equation that govern the dynamics of emitted particle. In fact, this method, usually called the Hamilton Jacobi method, is an extension of the complex path analysis firstly proposed by Padmanabhan et al. 0] However, all of these approaches of tunneling formalism to derive the Hawking radiation used the basic fact that the tunneling probability for the classically forbidden trajectory from inside to outside the horizon of black hole is given by the formula ( Γ = exp ) ImI, () where I is the classical action of the trajectory. So, it is clear that the essential thing in tunneling formalism is to calculate the imaginary part of classical action of emitted particle. The difference between these methods consists in how the classical action is calculated. However, the two methods are only applied to study the tunneling of scalar particles. Soon later, Kerner and Mann investigated the tunneling of spin-(/) fermions from the Schwarzschild black hole ] and the Kerr Newman black hole, ] which has been generalized to study other black hole backgrounds. 36] Yale and Mann 7] also studied the spin-(3/) gravitinos tunneling process along the classically forbidden trajectory from the event horizon of black hole. Chen et al. 8] studied the quantum gravity correction to the fermion tunneling spectrum based on the generalized uncertainty principle. One can refer to Refs. 9 0] for the recent comprehensive reviews on the topic of quantum tunneling formalism of Hawking radiation. In fact, a black hole can also radiate the vector particles, i.e. spin- bosons, at the Hawking temperature. The true emission spectrum should contain contribution of these spin- particles. The well-known massive vector particles are W ± and Z 0 bosons in Standard Model of particle physics, which are the three carriers of the weak Supported by National Natural Science Foundation of China under Grant No liran@htu.edu.cn c 06 Chinese Physical Society and IOP Publishing Ltd

2 78 Communications in Theoretical Physics Vol. 66 interaction. The dynamics of massive vector particles can be effectively described by the Proca field equation. More recently, Kruglov ] considered the tunneling radiation of spin- massive vector particles from black holes. By applying the WKB approximation to the Proca equation in the fixed background, one can study the quantum tunneling process of massive vector particles from the black hole. Along this line, many people investigates the tunneling of massive vector particles from the three-dimensional black holes, 3] Lorentzian wormholes, 4] non-stationary black holes, 5] Kerr Newmann black hole, 6] high dimensional black holes, 7] and the rotating black hole in conformal gravity. 8] It is shown that the massive vector particles tunneling method can be used to derive the Hawking temperatures of these black holes correctly. More recently, the tunneling processes of massive spin- particles from Reissner Nordstrom and Kerr black holes are investigated by taking the generalized uncertainty principle s influence into account. 9] In this paper, we will perform an analysis of quantum tunneling process of massive vector particles from the static spherically symmetric dilatonic black holes with arbitrarily coupling constant α, and from the rotating Kaluza Klein (α = 3), as well as the Kerr Sen (α = ) black holes. These dilatonic black hole solutions are obtained from the low energy effective field theory of string theory, which have qualitatively different properties from those in Einstein gravity. Our results show that the vector particles tunneling method is very robust to derive the Hawking temperatures of black holes. The remainders of this paper is arranged as follows. In Sec., we will give a derivation of Hawking radiation of vector particles from the static spherical symmetric dilaton black hole. In Sec. 3, the method is generalized to reproduce the Hawking temperature of the rotating Kaluza Klein black hole. Section 4 is devoted to once again check the validity of massive vector particles tunnelling method for the rotating Kerr Sen black hole. Final remarks are made in the last Section. Massive Vector Particles Tunneling from Spherical Symmetric Dilaton Black Hole The action for the dilaton gravity describing the dilaton field coupled to the U() gauge field in (3+)- dimensional is subject to the form S = dx 4 gr µ Φ µ Φ e αφ F µν F µν ], () 6π where Φ is dilaton field and F µν = µ A ν] is the U() gauge field respectively, with an arbitrary coupling constant α. The static spherically symmetric solution for the underlying theory can be expressed as 303] ds = R dt + R dr + r (dθ + sin θdφ ), Φ = α ( + α ln r r ), F = Q dt dr, (3) r where ( = (r r + )(r r ), R = r r ) α /(+α ), r in which the outer and inner horizons are respectively given by r ± = + α ± α M ± ] M ( α )Q. (4) The temperature of this black hole which can be calculated by surface gravity of event horizon in the standard formalism is given as T = ( r ) (α )/(+α ). (5) 4πr + r + For the case of the dilatonic coupling constant α 0, the inner horizon r = r is the curvature singularity. In the extreme limit r + = r, i.e. ( + α )M = Q, the inner horizon and the outer horizon coincide with each other, and the area of black hole horizon vanishes. In this paper, we only consider the nonextremal black hole case. Now, we focus on studying the massive vector particles tunneling radiation from the the static spherically symmetric dilatonic black hole. It is well known that the dynamics of massive vector particles are govern by the Proca field equation where D µ Ψ µν + m Ψν = 0, (6) Ψ µν = D µ Ψ ν D ν Ψ µ = µ Ψ ν ν Ψ µ, (7) with Ψ µ being the vector field in spacetime and m being the mass of the vector field Ψ µ. By using the antisymmetry of tensor field Ψ µν, the Proca equation can be reduced to the form of g µ ( gψ µν ) + m Ψν = 0. (8) According to the WKB approximation, the ansatz for the vector field Ψ µ is of the form i ] Ψ µ = C µ exp S(t, r, θ, φ). (9) By substituting this ansatz of Ψ µ into the Proca equation and keeping only the leading order in, we get the following equation g µλ g νρ (C ρ µ S λ S + C λ µ S ρ S) + m g νλ C λ = 0. (0) By substituting the metric (3) into the above equation, we can get g tt (g rr ( r S) g θθ ( θ S) g φφ ( φ S) + m )C 0 + g rr g tt t S r SC + g θθ g tt t S θ SC + g tt g φφ t S φ SC 3 = 0, () g tt t S r SC 0 + (g tt ( t S) g θθ ( θ S) g φφ ( φ S) + m )C + g θθ r S θ SC + g φφ r S θ SC 3 = 0, ()

3 No. Communications in Theoretical Physics 79 g tt t S θ SC 0 + g rr r S θ SC + (g tt ( t S) g rr ( r S) g φφ ( φ S) + m )C + g φφ θ S φ SC 3 = 0, (3) g tt g φφ t S φ SC 0 + g rr g φφ r S φ SC + g θθ g φφ θ S φ SC + g φφ (g tt ( t S) g rr ( r S) g θθ ( θ S) + m )C 3 = 0.(4) Considering the symmetry of the spacetime, we can carry out separation of variables as the following S = ωt + W(r) + jφ + Θ(θ), (5) where ω and j denote the energy and angular momentum of the emitted vector particle respectively. Then we have g tt (g rr W g θθ J θ gφφ j + m )C 0 g rr g tt ωw C g θθ g tt ωj θ C g tt g φφ ωjc 3 = 0, (6) g tt ωw C 0 + (g tt ω g θθ J θ g φφ j + m )C + g θθ W J θ C + g φφ W J θ C 3 = 0, (7) g tt ωj θ C 0 + g rr W J θ C + (g tt ω g rr W g φφ j + m )C + g φφ J θ jc 3 = 0, (8) g tt g φφ ωjc 0 + g rr g φφ W jc + g θθ g φφ J θ jc + g φφ (g tt ω g rr W g θθ J θ + m )C 3 = 0, (9) where J θ denotes θ Θ. One can treat these equations as a matrix equation K(C 0, C, C, C 3 ) T = 0. These four equations have non-trivial solution for (C 0, C, C, C 3 ) if and only if the determinant of the coefficient matrix K should be equal to zero. The determinant of the coefficient matrix is then given by det K = m (j + sin θjθ ) + sin θ(w ω R 4 m R )] 3 R 6 4 sin 8. (0) θ By solving detk = 0, we obtain ω R 4 + m R + (J W θ = ± + j / sin θ). () As discussed in the Hamilton Jacobi method, 3334] one solution corresponds Dirac particles moving away from the outer event horizon and the other solution corresponds the particles moving toward the outer event horizon. So we have ω R 4 + m R + (Jθ W ± (r) = ± + j / sin θ) dr, () where W ± denote the radial action of the outgoing particles and the ingoing particles, respectively. Integrating around the pole at the horizon r = r +, one can get the imaginary part of W ± ImW ± (r) = ± πωr(r +) r + r ( = ±πωr + r ) (α )/(α +). (3) r + The WKB approximation tells us that the tunneling probabilities of particles along the classically forbidden trajectory are related to the imaginary part of action. More explicitly, the probabilities of crossing the outer horizon each way are respectively given by P out = exp ] ImS = exp ] (ImW + + ImΘ), P in = exp ] ImS = exp ] (ImW + ImΘ). (4) So the probability of a massive vector particle tunneling from inside to outside the horizon is finally given by Γ = P out = exp 4 ] P in ImW + = exp 4 ( πωr + r ) (α )/(α +)]. (5) r + From the emission probability (5), the Hawking radiation spectrum of massive vector particles from the nonrotating linear dilaton black hole can be deduced following the standard arguments,35] N(ω) = e ω/th, (6) with the Hawking temperature of the spherical symmetric dilaton black hole as T H = ( r ) (α )/(+α ), (7) 4πr + r + where we have set =. This result calculated using the massive vector particles tunneling formalism is consistent with result calculated by using gravitational anomaly method in Ref. 36], fermion tunneling method, 37] and the Hamilton Jacobi method in Ref. 38]. It should be noted that we have not considered the couplings between the massive vector field Ψ ν and the background dilatonic field Φ and gauge field F. Additionally, the tunneling rate (5) is derived by neglecting the higher terms about ω, and the resulting spectrum is purely thermal black body spectrum. If taking the energy conservation and self-gravitation effect into account, the higher terms will be present in the tunneling rate, and the radiation spectrum is no longer thermal. In the next two sections, to further verify the validity of application of massive vector particles tunneling method to

4 80 Communications in Theoretical Physics Vol. 66 the dilatonic black holes, we additionally take the rotating Kaluza Klein black hole and Kerr Sen black hole as examples to discuss the Hawking radiations of bosons. 3 Massive Vector Particles Tunneling from Rotating Kaluza Klein Black Hole In this section, we will calculate the massive vector particles tunneling process from the rotating Kaluza Klein black hole. The Kaluza Klein black hole is an exact solution of the dilatonic action () with the coupling constant α = 3. It is derived by a dimensional reduction of the boosted five-dimensional Kerr solution to four dimensions. The metric is given by 3940] ds = Z B dt az sin θ B ν + B(r + a ) + a sin θ Z B B dtdφ + dr + Bdθ ] sin θdφ, (8) where the metric functions are given by Z = µr, B = + ν Z ν, (9) = r + a cos θ, = r µr + a, (30) and µ, ν and a are parameters related to the mass, boost and specific angular momentum respectively. The dilatonic field and the gauge potential are given by Φ = 3/ lnb, ν Z A = ( ν ) B dt aν sin θ Z dφ. (3) ν B The physical mass M, the charge Q, and the angular momentum J are expressed by ν ] M = µ + ( ν, ) (3) Q = µν ν, (33) µa J =. ν (34) When ν = 0, the solution is reduced to the Kerr black hole. The outer and inner horizons, which are determined by the equation = 0 are respectively given by r ± = µ ± µ a, (35) thus the regular horizon exists if µ a. The temperature and the the angular velocity of this black hole are given as ν µ T = a π(r+ +, Ω H = a ν a ) r+. (36) + a For the convenience of later computation, we also present the nonzero components of the inverse metric, which are given by g tt = B(r + a ) + a sin θ(z/b), g rr = B, g θθ = B, gφφ = Z Bsin θ, az gtφ = B ν. (37) Now, we want to employ the method used in the last section to study the massive vector particles tunneling radiation in the rotating Kaluza Klein black hole. By inserting the metric (8) into the Proca equation (6), ignoring the higher order of terms of, and taking the variables separation of the action (5), we can finally get the following equations after some algebra g tt (g rr W g θθ J θ gφφ j + m ) + (g tφ ) j ]C 0 + g rr (g tt ωw + g tφ W j)c + g θθ (g tt ωj θ + g tφ J θ j)c + g tφ (g rr W g θθ J θ + g tφ ωj + m ) g tt g φφ ωj]c 3 = 0, (38) (g tt ωw + g tφ W j)c 0 + (g tt ω g θθ J θ gφφ j + g tφ ωj + m )C + (g θθ W J θ )C + (g tφ ωw + g φφ W J θ )C 3 = 0, (39) (g tt ωj θ + g tφ J θ j)c 0 + (g rr W J θ )C + (g tt ω g rr W g φφ j + g tφ ωj + m )C + (g tφ ωj θ + g φφ J θ j)c 3 = 0, (40) g tφ (g rr W g θθ J θ + g tφ ωj + m ) g tt g φφ ωj]c 0 + g rr (g tφ ωw + g φφ W j)c + g θθ (g tφ ωj θ + g φφ J θ j)c + g φφ (g tt ω g rr W g θθ J θ + m ) + (g tφ ) ω ]C 3 = 0. (4) Once again, these four equations have non-trivial solution for (C 0, C, C, C 3 ) if and only if the determinant of the coefficient matrix K should be equal to zero. The determinant of the coefficient matrix is then given by By solving detk = 0, we obtain detk = m (g tt g φφ g tφ g tφ )(g tt ω ωjg tφ + g φφ j m + g rr W + g θθ J θ) 3. (4) g W tt ω = ± + ωjg tφ g φφ j + m g θθ Jθ. (43) g rr By substituting the inverse metric (37) into the above equation, we can get W ± (r) = ± (B (r + a ) + a sin θz)ω ωjaz j ( Z) ν sin + F dr, (44) θ

5 No. Communications in Theoretical Physics 8 with F = Bm J θ, (45) where W ± denote the radial action of the outgoing particles and the ingoing particles, respectively. Integrating around the pole at the horizon r = r + and applying the residue theorem in complex analysis, one can get the imaginary part of W ± ImW ± (r) = ± π(r + + a )(ω jω H ) ν µ a. (46) It should be noted that although the integrand in (44) is a function of the angular coordinate θ, the result (46) is not relevant to the angular coordinate θ as expected. The probability of a massive vector particle tunneling from inside to outside the horizon is finally given by Γ = P out = exp4imw + ] P in = exp π(r + + a )(ω jω H )]. (47) ν µ a From the emission probability (47), the Hawking radiation spectrum of massive vector particles from the rotating Kaluza Klein black hole can be deduced following the standard arguments,,35] N(ω) = e (ωjω)/th, (48) which is just the Planck distribution including the chemical potential for the azimuthal angular quantum number j with the Hawking temperature of the rotating Kaluza Klein black hole as ν µ T H = a π(r+ + a. (49) ) So once again, we have reproduced the Hawking temperature of the Kaluza Klein black hole by adopting the massive vector particles tunneling method. This result is also consistent with the result calculated by using other methods. 3638] In the above process, the higher terms of ω and j are also neglected in deriving the tunneling probability (47). The thermal spectrum (48) will be corrected if considering the energy and angular momentum conservation. It shows that the massive vector particles tunneling method is very robust in studying the Hawking radiation of bath the nonrotating and the rotating dilatonic black holes. In the next section, we will further apply this method to study the tunneling radiation of vector particles from the Kerr Sen black hole. 4 Massive Vector Particles Tunneling from Kerr Sen Black Hole In this section, we will continue to calculate the tunneling process of massive vector particles from the rotating Kerr Sen black hole. 4] The Kerr Sen black hole is a solution to the bosonic part action of the low energy limit of heterotic superstring theory. The action contains the interaction of dilaton field Φ, the axion field H, and the U() gauge field F, in which the coupling constant α between the dilaton and the U() gauge field is set to unity. The action is then given by S = 6π dx 4 g R µ φ µ φ e φ F µν F µν e4φ H ]. (50) Sen adopted the solution generating method to obtain a new solution from the uncharged Kerr solution. The metric is given by ds = a sin θ dt 4µra cosh β sin θ dtdφ + dr + dθ + Λ sin θdφ, (5) where the metric functions,, and Λ are defined as = r µr + a, = r + a cos θ + µr sinh β, Λ = (r + a )(r + a cos θ) + µra sin θ + 4µr(r + a )sinh β + 4µ r sinh 4 β. (5) The dilaton field, axion field and the U() gauge potential are respectively given by φ = ln r + a cos θ, B tφ = a sin θ µr sinh β, A t = µr sinh β, A φ = a sin µr sinh β θ. (53) The mass M, the charge Q, and the angular momentum J of the Kerr Sen black hole are given by the parameters µ, β, and a as M = µ ( + coshβ), Q = µ sinh β, J = aµ ( + cosh β). (54) The outer and inner horizons are given by the same equation as Eq. (33). The temperature and the the angular velocity of this black hole are given as µ a T = π(r+ + a )cosh β, (55) Ω H = a (r + + a )cosh β. (56) Now, we want to further check the validity of massive vector particles tunneling method in deriving the Hawking temperature of the rotating Kerr Sen black hole. We adopt the same procedure presented in the last section. By inserting the metric (5) into the Proca equation (6), ignoring the higher order of terms of, and taking the variables separation of the action (5), we can easily get the same equations in Eqs. (38) (4) after some algebra. Once again, these four equations have non-trivial solution for (C 0, C, C, C 3 ) if and only if the determinant of the

6 8 Communications in Theoretical Physics Vol. 66 coefficient matrix K should be equal to zero. The determinant of the coefficient matrix is also given by Eq. (4). By solving det K = 0 and substituting the inverse metric into the result, we can finally arrive at the following equation this time W = ± ω Λ/ ωj(µra cosh β)/ j ( a sin θ)/sin θ + (m J θ /) /. (57) So we get W ± (r) = ± ω Λ 4ωjµra cosh β j a sin θ ( ) sin + m J θ dr, (58) θ where W ± denote the radial action of the outgoing particles and the ingoing particles, respectively. Integrating around the pole at the horizon r = r +, one can get the imaginary part of W ± where ImW ± (r) = ± πµr + cosh β(ω jω H ) µ a, (59) a Ω H = µr + cosh β is just the angular momentum of Kerr Sen black hole. The probability of a massive vector particle tunneling from inside the horizon to outside is given by Γ = P out = exp4imw + ] P in = exp 4πµr + cosh β(ω jω H ) ]. (60) µ a From the emission probability (59), the Hawking radiation spectrum of massive vector particles from the rotating Kerr Sen black hole can be deduced following the standard arguments,35] N(ω) = e (ωjωh)/th, (6) with the Hawking temperature of the rotating Kerr Sen black hole as µ a T H = π(r+ + a )cosh β. (6) This result calculated by the massive vector particles tunneling formalism is also consistent with result calculated using other methods. 3638] Now, we can conclude that both Boson and fermion can be emitted from the Kerr Sen black hole in the thermal spectrum with the same Hawking temperature. 5 Conclusion In this paper, we have investigated the massive vector particles Hawking radiation from the static spherical symmetric dilatonic black hole, the rotating Kaluza Klein black hole, and the rotating Kerr Sen black hole by adopting the tunneling formalism. By applying the WKB approximation to the field equation of massive vector bosons, we have successfully derived the tunneling probabilities and radiation spectrums of the emitted vector particles from these dilatonic black holes. The Hawking temperatures of these dilatonic black holes have been recovered, which is consistent with the previous results in the literature. This means that the vector particles tunneling method is very robust in studying the Hawking radiation of dilatonic black holes. It should be emphasized that we have neglected the higher terms of in the deriving process. The results show that these dilatonic black holes emit massive vector particles in the form of black body spectrums at the Hawking temperatures. It is expected that the energy and angular momentum conservation can be restored by taking the higher terms in into account, where the resulted spectrum will not thermal any longer. This will be addressed in the near future. Acknowledgments Ran Li is also supported by the Foundation for Young Key Teacher of Henan Normal University. References ] S.W. Hawking, Commun. Math. Phys. 43 (975) 99. ] T. Damour and R. Ruffini, Phys. Rew. D 4 (976) 33. 3] S. Christensen and S. Fulling, Phys. Rev. D 5 (977) ] S.P. Robinson and F. Wilczek, Phys. Rev. Lett. 95 (005) ] S. Iso, H. Umetsu, and F. Wilczek, Phys. Rev. Lett. 96 (006) ] M.K. Parikh and F. Wilczek, Phys. Rev. Lett. 85 (000) ] J. Zhang and Z. Zhao, J. High Energy Phys. 0 (005) ] J. Zhang and Z. Zhao, Phys. Lett. B 68 (005) 4; J. Zhang and Z. Zhao, Phys. Lett. B 638 (006) 0.

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