Hawking radiation from Kerr Newman Kasuya black hole via quantum anomalies

Transcription

1 Vol 17 No 6, June 008 c 008 Chin. Phys. Soc /008/1706/31-05 Chinese Physics B and IOP Publishing Ltd Hawking adiation fom Ke Newman Kasuya black hole via quantum anomalies He Tang-Mei a, Fan Jun-Hui b, and Wang Yong-Jiu c a Laboatoy Cente, Guangzhou Univesity, Guangzhou , China b Cente fo Astophysics, Guangzhou Univesity, Guangzhou , China c Institute of Physics, Hunan Nomal Univesity, Changsha , China Received 9 Apil 007; evised manuscipt eceived 3 Decembe 007 We have studied the Hawking adiation of the Ke Newman Kasuya black hole via gauge and gavitational anomaly in the dagging coodinates. The fluxes of the electomagnetic cuent and the enegy momentum tenso fo each patial wave in two-dimensional field ae obtained. Keywods: Hawking adiation, Ke Newman Kasuya black hole, anomaly PACC: 9760L, 0460K, 0470D 1. Intoduction The study of Hawking adiation has attacted moe and moe attention in theoetical physics since Hawking [1] fist poved the existence of black hole adiation, especially in ecent yeas. Thee ae seveal deivations of Hawking adiation in the liteatue. One of them was poposed by Paikh and Wilczek seveal yeas ago. [] They teated Hawking adiation as a tunnelling pocess, by using the Wentzel Kames Billouin WKB method. Following thei famewok, Hawking adiations fom seveal types of black holes have been studied. [3 16] Vey ecently, Robinson and Wilczek [17] have pesented a new patial deivation of Hawking adiation, which ties its existence to the cancellation of gavitational anomaly at the hoizon of the Schwazschildtype black hole. The anomaly in field theoy is defined if the symmety of the action o the coesponding consevation law, valid in the classical theoy, is violated in the quantum theoy. The basic idea of Robinson and Wilczek s wok is as follows: by educing dimensions, the physics nea the hoizon can be descibed with an infinite collection of fee 1+1- dimensional fields because the mass and the inteaction tems of quantum field in the backgound ae suppessed. If only the effective field theoy nea the hoizon is consideed, it becomes chial since outgoing modes thee ae eliminated. And quantum mechanically, the effective action fo the metic due to matte fields becomes anomalous with espect to geneal coodinate tansfomations. The Hawking flux of enegy momentum tenso is detemined so that it cancels the gavitational anomaly in the consistent fom at the hoizon. Robinson and Wilczek s appoach is effectively equivalent to dilaton-coupled tace anomaly in two dimensions. With egad to this, many effots have been devoted to extending thei poposal to many othe cases. [18 5] Howeve, much less attention was paid to the case of Hawking adiation of the Ke Newman Kasuya black hole, which is a type of otating black hole with electic chage and magnetic chage. Hawking flux fom this type of black hole can also be detemined in tems of the values of anomalies at the hoizon by demanding gauge invaiance and geneal coodinate covaiance at the quantum level. The emainde of this pape is oganized as follows. In Section, we investigate the Hawking adiation fom the Ke Newman Kasuya black hole in the Boye Lindquist coodinates and in the dagging coodinates. In Section 3, we give some discussion and conclusions. Poject suppoted by the National Natual Science Foundation of China Gant Nos and and the State Key Pogam fo Basic Reseach of China Gant No 003CB Coesponding autho. wyj@hunnu.edu.cn

2 3 He Tang-Mei et al Vol. 17. Hawking adiation fom Ke Newman Kasuya black hole via quantum anomalies Accoding to an exact solution of a otating dyon black hole, the metic of the fou-dimensional Ke Newman Kasuya black hole in the Boye Lindquist coodinates can be expessed as [6,7] ds = 1 M e q dt + Σ Σ d M e + Σdθ q a sin θ + Σ + + a sin θdφ a sin θ Σ M e q dtdφ, 1 whee Σ = + a cos θ, = + a + e + q M = +, a = J M. The paametes M, e, q epesent the mass, the electic chage and the magnetic chage of the black hole, espectively, and a is the angula momentum pe unit mass of the black hole. +, ae the oute and the inne hoizons, ± = M ± M a e q. By using a dimensional eduction technique, the Ke Newman Kasuya black hole in the Boye Lindquist coodinates t,, θ, φ can be educed to that in a two-dimensional spacetime. Fo a while we conside a scala field fo simplicity. The action fo the scala field in the Ke Newman Kasuya spacetime is S[ϕ] = 1 d 4 x gϕ ϕ + S int = 1 [ a sin θ + a dtddθdφ sin θϕ t sin θ θ sin θ θ + a sin θ sin θ φ + a a t φ ]ϕ + S int, whee S int includes a mass tem, potential tems and inteaction tems. Taking the limit +, and consideing that the kinetic tem makes a dominant contibution to the action, we can ignoe the mass and inteaction tems S int. Keeping the dominant tems, the action becomes S[ϕ]= 1 a [ dtddθdφ sin θϕ + + a t + φ + a + + a t φ ]ϕ. 3 Then, we intoduce the dagging coodinates ξ,, θ, ψ though ξ = t and ψ = φ Ω + t, whee Ω + is the dagging speed at the oute hoizon, and Ω + g tφ a g φφ = a.[8,9] To conside the physics nea the hoizon, we take the new dagging coodinates and tansfom them into the totoise coodinate defined by d d = + + a, then the action + expession 3 can be ewitten as S[ϕ] = a dξddθdψ sin θϕ Ω + 1 f ξ + f ϕ. 4 The expession shows that the angula deivative tems disappea completely fom the action. Pefoming the patial wave decomposition of the scala field in the Ke Newman Kasuya black hole in tems of the spheical hamonics ϕ = l,m ϕ lm t, Y lm θ, φ at the hoizon, the action finally eads S[ϕ] = a 1 dξdϕ lm Ω + l,m 1 f ξ + f ϕ lm, 5 whee ϕ lm is consideed as a 1+1- dimensional complex scala field in the backgound of the dilaton Φ, metic g µν and U1 gauge field A µ. Because

3 No. 6 Hawking adiation fom Ke Newman Kasuya black hole via quantum anomalies 33 of the static backgound, howeve, the contibution of the dilaton backgound to the total flux can be neglected. Then, we can obtain the effective twodimensional metic fom the above action expession as ds = fdξ + 1 f d. So we have g ξξ = f = + a, g = 1 f, A e + q t = + a, and A = 0. Making the totoise coodinate tansfomation and then pefoming the potential wave decomposition, we find that the effective adial potential fo patial wave modes of the field ll + 1/ contains the facto f and vanishes exponentially fast nea the hoizon. The same consideation may be applied to mass tems and othe inteactions. Nea the hoizon of the Ke Newman Kasuya black hole, each patial wave of scala fields behaves as a chaged black hole. These two-dimensional fields ae inteacting with each othe but because of the axial symmety of the Ke Newman Kasuya black hole metic in the φ-diection, the azimuthal quantum numbe, m of Y lm, is conseved. In the famewok of Robinson and Wilczek s appoach, once the ingoing modes fall into the black hole, they neve come out classically and cannot affect the physics outside the black hole. Quantum mechanically, howeve, if we neglect the ingoing modes nea the hoizon, the effective two-dimensional theoy becomes chial nea the hoizon, and the gauge symmety and the geneal coodinate covaiance become anomalous due to the gauge and gavitational anomalies. In the case of Ke Newman Kasuya black hole, by using the effective two-dimensional eduction, each patial wave exhibits an oiginal gauge symmety with espect to the electomagnetic field of the black hole and an induced U1 gauge symmety oiginating fom the isomety along the φ diection. So the effective chial fo Ke Newman Kasuya black hole contains two gauge anomalies and a gavitational anomaly in all afte omitting the classically ielevant ingoing modes at the hoizon. If we intoduce the dagging coodinate system, howeve, the U1 gauge anomaly associated with the induced symmety oiginating fom the isomety along the φ-diection vanishes. An obseve est in the dagging coodinates would not obseve the U1 gauge cuent flux, fo he is co-otating with the black hole. Theefoe, the effective theoy contains only one gauge anomaly associated with the electomagnetic field of the black hole. Fistly, we investigate the flux fom the U1 gauge anomaly. We divide the two-dimensional egion into two ones, i.e. the outside of the hoizon egion + + ε and the nea-hoizon egion ε. Afte omitting the ingoing modes in the egion ε, the cuent exhibits a gauge anomaly thee. Take the ε 0 ultimately, the explicit fom of the gauge anomaly will be given as [0] µ J µ = m 4π g εµν µ A ν, 6 whee ε µν is the two-dimensional Levi-Civita tenso, the convention ε 01 =+1 has been used and m has been teated as the gauge chage of the U1 gauge field A ν. Since the cuent J µ tansfoms non-covaiantly, we should define a new covaiant cuent J µ = J µ m 4π g ελν A λ, which satisfies µ J µ = m 4π g ε µνf µν, whee the coefficient of the covaiant anomaly is twice as lage as that of the consistent anomaly. In the outside of the hoizon egion + + ε, since thee exists no anomaly as shown by the fundamental theoy, the electomagnetic cuent is conseved, and it satisfies the conseved equation JO = 0. 7 While in the nea-hoizon egion ε, since thee exists only an outgoing field, the cuent satisfies the anomalous equation J H = m 4π A t. 8 The subscipts O and H in Eqs.7 and 8 epesent the values in the egions + + ε and ε, espectively. Solve the above equations, and we will have and J O = c O, 9 J H = c H + m 4π A t A t +, 10 whee c O and c H ae integation constants and epesent the values of the consistent cuent at infinity and the hoizon, espectively. Since the cuent J µ should be the sum of the both pats JO and J H, we have J µ = J µ O Θ H + J µ H H, 11

4 34 He Tang-Mei et al Vol. 17 whee Θ H = Θ + ε is a scala step function and H = 1 Θ H is a scala top hat function. Unde gauge tansfomations, the vaiation of the effective action nea the hoizon is given by δw = d x g λ µ J µ, whee λ is a gauge paamete. Then by using the anomaly equation, the vaiation becomes δw = d x [ g λ δ + ε JO J H ] + m m 4π A t + 4π A th. 1 The total effective action must be gauge invaiant, so we have δw =0. Since the last tem cannot be cancelled by delta-function tems, it should be cancelled by quantum effects of the classically ielevant ingoing modes. Thus, we havej O J H + m 4π A t = 0, and substituting the expessions 9 and 10 to it yields c O = c H m 4π A t In ode to detemine the cuent flux, we need to fix the value of the cuent at the hoizon c H. Since the condition should be gauge covaiant, we impose that the coefficient of the covaiant cuent at the hoizon should vanish. Since J = J + m 4π A th, and J + = 0, those conditions detemine the value of the chage flux to be c O = m π A t + = m e + q + π+ + a, 14 which is exactly equal to that of blackbody adiation at the Hawking tempeatue with an appopiate chemical potential. Then, we investigate the flux of the enegy momentum tenso. In the egion + + ε, thee exists an effective gauge potential. Thus the enegy momentum tenso satisfies the modified consevation equation T to = J O A t, 15 which is elated to a componenttt of a 4-dimensional enegy momentum tenso. In the nea-hoizon egion ε, the enegy momentum tenso exhibits a gavitational anomaly. The explicit fom of the gavitational anomaly is T th = F tj H + A t µ J µ H + N t, 16 whee F t = A t, and N t = 1 19π f + ff. The second tem comes fom the gauge anomaly and the thid is the gavitational anomaly fo the consistent enegy momentum tenso. The fist and second tems can be combined in tems of the covaiant cuent J H into F J H. Using the esults we have obtained befoe and solving Eqs.15 and 16, we have TtO = a O + c O A t, 17 TtH = a H + d c O A t + m + 4π A t + Nt, 18 whee a O and a H ae integated constants, the values of the enegy flux at the infinity and the hoizon, espectively. The enegy momentum tenso should be the sum of enegy momentum tensos in the two egions, i.e. T ν µ = T µ νo Θ H + T µ νh H. Unde the geneal coodinate tansfomation x µ x µ λ µ, the vaiation of the effective action is induced as δ λ W = d x gλ t µ T µ t = d xλ [δ t + ε TtO T th + m 4π A t + Nt + c O A t + m 4π A t + N t ], 19 whee the second tem comes fom the classical effect of the backgound electomagnetic field fo constant flow, and the thid should be cancelled by the quantum effect of the ingoing modes. In ode to estoe the diffeomophism covaiance at the hoizon, the coefficient of the fist tem should vanish. So we have a O = a H + m 4π A t + N t +, 0 in which a O = TtO c OA t. Imposing a condition [19,0,30] unde which the covaiant enegy momentum tenso T t = Tt + Nt is equied to vanish at the hoizon, we can obtain a H = N t + = κ 4π, 1 whee κ = 1 f + is the suface gavity of the black hole. The flux of the enegy momentum tenso, which is equied to estoe the geneal coodinate covaiance at a quantum level in the effective field theoy, is a O = m e + q 4π+ + Nt + = m e + q 4π+ + π 1 T +,

5 No. 6 Hawking adiation fom Ke Newman Kasuya black hole via quantum anomalies 35 whee T + = κ is the Hawking tempeatue of the π black hole at the hoizon. 3. Discussion and conclusions The famewok of this pape, howeve, is a little diffeent fom that of Robinson and Wilczek, in which the outgoing modes nea the hoizon ae eliminated and thus effective theoy is chial thee, and the Hawking flux of enegy-momentum tenso is detemined such that it cancels the gavitational anomaly in the consistent fom at the hoizon. In this pape, ingoing modes, which ae classically ielevant to physics outside the hoizon, ae integated out nea the hoizon. The Hawking fluxes ae detemined by equiing the covaiant cuent and the enegy momentum tenso to vanish at the hoizon. We now compae ou esults with the fluxes fom the blackbody adiation moving in the positive - diection at an invese tempeatue β = 1 with a chemical potential. T + In the dagging coodinates, the Hawking distibution is given by Planck distibution N Q,M ω = 1, whee ω = ω mω + is the e βω e +q A t enegy caied by the obseve in the dagging coodinate system. Hee we have used the Planck distibution fo femions in ode to avoid the poblem of supeadiance which is elated to scatteings away fom the hoizon. So the electomagnetic cuent and enegy momentum tenso fluxes of Hawking adiation ae given by F Q =m F M = 0 1 π [N +Qω N Q ω ]dω = m e + q + π+ + a, 3 0 ω π [N +Qω + N Q ω ]dω = m e + q 4π + + π 1 T +. 4 The esults show that in the dagging coodinates the electomagnetic cuent and the enegy tenso fluxes of Hawking adiation fom Ke Newman Kasuya black hole have the equivalent foms to those obtained by equiing the gauge invaiance and the geneal coodinate covaiance at a quantum level to hold in the effective theoy. Refeences [1] Hawking S W and Page D N 1983 Math. Phys [] Paikh M K and Wilczek F 000 Phys. Rev. Lett [3] Cai X, Wu S Q, Yan M L and Zeng Y 003 Acta Phys. Sin in Chinese [4] Zhang J Y and Zhao Z 003 Acta Phys. Sin in Chinese [5] Yang S Z 004 Acta Phys. Sin in Chinese [6] Jiang Q Q, Li H L and Yang S Z 005 Chin. Phys [7] Jiang Q Q, Li H L and Yang S Z 005 Chin. Phys [8] Meng Q M 005 Acta Phys. Sin in Chinese [9] Han Y W 005 Acta Phys. Sin in Chinese [10] Jiang Q Q, Li H L and Yang S Z 006 Acta Phys. Sin in Chinese [11] Zhang J Y and Zhao Z 006 Acta Phys. Sin in Chinese [1] Jiang Q Q and Wu S Q 006 Acta Phys. Sin in Chinese [13] Hu Y P, Zhang J Y and Zhao Z 007 Acta Phys. Sin in Chinese [14] Cai X, Jiang Q Q and Wu S Q 007 Acta Phys. Sin in Chinese [15] Jiang J J, Meng Q M and Su J Q 007 Acta Phys. Sin in Chinese [16] Jiang J J, Meng Q M and Su J Q 007 Acta Phys. Sin in Chinese [17] Robinson S P and Wilczek F 005 Phys. Rev. Lett [18] Iso S, Umetsu H and Wilczek F 006 Phys. Rev. Lett [19] Muata K and Soda J 006 Phys. Rev. D [0] Iso S, Umetsu H and Wilczek F 006 Phys. Rev. D [1] Vagenas E C and Das S 006 JHEP , hepth/ [] Setae M R 006 Eu. Phys. J. C [3] Iso S, Moita T and Umetsu H 006 hep-th/06186 [4] Xu Z and Chen B 006 Phys. Rev. D [5] Jiang Q Q and Wu S Q 007 Phys. Lett. B [6] Kasuya M 198 Phys. Rev. D [7] Wang Y J and Tang Z M 1990 Theoy and Effects of Gavitation Changsha: Pess of Hunan Science and Technology p9 [8] He T M and Wang Y J 006 Chin. Phys [9] Li A and Wang Y J 006 ApJ Lett [30] Hu P H and Wang Y J 006 Chin. Phys