Massive complex scalar field in the Kerr Sen geometry: Exact solution of wave equation and Hawking radiation

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1 JOURNAL OF MATHEMATICAL PHYSICS VOLUME 44, NUMBER 3 MARCH 2003 Massive compex scaar fied in the Kerr Sen geometry: Exact soution of wave equation and Hawking radiation S. Q. Wu a) Interdiscipinary Center for Theoretica Study, and Department of Astronomy and Appied Physics, University of Science and Technoogy, Hefei , Peope s Repubic of China and Institute of Partice Physics, Hua-Zhong Norma University, Wuhan , Peope s Repubic of China X. Cai b) Institute of Partice Physics, Hua-Zhong Norma University, Wuhan , Peope s Repubic of China Received 4 September 2001; accepted 2 December 2002 The separated radia part of a massive compex scaar wave equation in the Kerr Sen geometry is shown to satisfy the generaized spheroida wave equation which is, in fact, a confuent Heun equation up to a mutipier. The Hawking evaporation of scaar partices in the Kerr Sen back hoe background is investigated by the Damour Ruffini Sannan method. It is shown that quantum therma effect of the Kerr Sen back hoe has the same character as that of the Kerr Newman back hoe American Institute of Physics. DOI: / I. INTRODUCTION In a recent paper, 1 we have investigated exact soution of a massive compex scaar fied equation in the Kerr Newman back hoe background, and demonstrated that both its radia part and its anguar part can be transformed into the form of a generaized spheroida wave equation. 2 Previous work on soution of a massive scaar wave equation in the Kerr Newman space time had been competed in Refs. 3 and 4. It is interesting to extend our anaysis to soution of a scaar wave equation in a Kerr Sen back hoe background. 5 The Kerr Sen soution arising in the ow energy effective string fied theory is a rotating charged back hoe generated from the Kerr soution. The thermodynamic property of this twisted Kerr back hoe was discussed in Ref. 6 by using separation of the Hamiton Jacobi equation of a test partice. The aim of this paper is to study some exact soutions to a massive charged scaar wave equation and to find its connection to the confuent Heun equation 7 as we as to investigate quantum therma effect of scaar partices on the Kerr Sen space time. The paper is organized as foows: In Sec. II, we separate a massive charged scaar fied equation on the Kerr Sen back hoe background into the radia and anguar parts. Section III is devoted to transforming the radia part into a generaized spheroida wave equation and to reating it to the confuent Heun equation. Then, we investigate quantum therma effect of scaar partices in the Kerr Sen space time in Sec. IV. Finay, we summarize our discussions in the concusion section. II. SEPARATING VARIABLES OF KLEIN GORDON EQUATION ON THE KERR SEN BLACK HOLE BACKGROUND Constructed from the charge neutra rotating Kerr back hoe soution, the Kerr Sen soution 5 is an exact cassica four dimensiona back hoe soution in the ow energy effective heterotic string fied theory. In the Boyer Lindquist coordinates, the Kerr Sen metric and the eectromagnetic fied vector potentia can be rewritten as 6 a Eectronic mai: sqwu@ustc.edu.cn b Eectronic mai: xcai@ccnu.edu.cn /2003/44(3)/1084/5/$ American Institute of Physics

2 J. Math. Phys., Vo. 44, No. 3, March 2003 Massive compex scaar fied in Kerr Sen geometry 1085 ds 2 dta sin2 d 2 sin2 adta2 sin 2 d d2 2 dr2, A Qr dta sin2 d, 1 where r 2 2(bM)ra 2 (rr )(rr ), r 2 2bra 2 cos 2 and r Mb with (Mb) 2 a 2. This metric describes a back hoe carrying mass M, charge Q, anguar momentum J Ma, and magnetic dipoe moment Qa. The twist parameter b is reated to the Sen s parameter via bq 2 /2MM tanh 2 (/2). Because Mb0, rr is a new singuarity in the region r 0, the event horizon of the Kerr Sen back hoe is ocated at rr. The area of the er event horizon of the twisted Kerr soution 5 is given by A 4(r 2 2br a 2 )8Mr. We consider the soution of a massive charged test scaar fied on the Kerr Sen back hoe background we use Panck unit system Gck B 1 through the paper. Because the Kerr Sen metric 1 ony differs from the Kerr Newman soution by the form of two functions and, the minima eectromagnetic couping Kein Gordon fied equation satisfied by the compex scaar wave function with mass and charge q in such a space time can be separated as (t,r,,)r(r)s m,0 (ka,)e i(mt), in which the anguar part S m,0 (ka,) is an ordinary spheroida anguar wave function with spin weight s0, whie the radia part can be given as foows: r r Rr Arma2 k 2 2DrRr0, 2 here is a separation constant, A2MqQ, DAM 2, k 2 2 we assume that ). For ater convenience, we aso denote BA(Mb)ma and introduce W (A B)/2. With further substitution R(r)(rr ) i(ab)/2 (rr ) i(ab)/2 F(r), we can transform Eq. 2 for R(r) into a modified generaized spheroida wave equation with imaginary spin weight ia and boost weight ib for F(r), 1 r 2 Fr2iB1iArMb r Frk 2 2DriAFr0. 3 Equation 2 has two reguar singuar points rr with indices iw and iw, respectivey, whereas Eq. 3 has indices 0, 2iW and 0, 2iW at two singuarities r r, respectivey. The infinity is an irreguar singuarity of Eqs. 2 and 3. Equation 3 has the same form as the radia part of the massive compex scaar wave equation in the Kerr Newman geometry 1 with its soution when 0) named as the generaized spheroida wave function. 2 It is interesting to note that a specia soution of function F(r) satisfies the Jacobi equation of imaginary index when qq/m namey, kd0). III. GENERALIZED SPHEROIDAL WAVE FUNCTION AND HEUN EQUATION In this section, we sha show that the generaized spheroida wave equation 3 of imaginary number order is, in fact, a confuent form of Heun equation. 7 To this end, et us make a coordinate transformation rmbz and substitute R(r)(z1) i(ab)/2 (z1) i(ab)/2 F(z) into Eq. 2, then we can reduce it to the foowing standard forms of a generaized spheroida wave equation: 1,7 z 2 1Rz2zRzk 2 z 2 12Dz AzB2 z 2 2DMbRz0 1 4

3 1086 J. Math. Phys., Vo. 44, No. 3, March 2003 S. Q. Wu and X. Cai and z 2 1Fz2iB1iAzFzk 2 z 2 1 2Dz2DMbiAFz0, 5 where a prime denotes the derivative with respect to its argument. The spin-weighted spheroida wave function F(z) is symmetric under the refect k k. Letting F(z)e ikz G(z) with oss of generaity, we can transform Eq. 5 to z 2 1Gz2iB1iAzikz 2 1Gz2ik1iAiD/kz2kBiA 2DMbGz0. 6 By means of changing variabe z12x, we arrange the singuarities rr (z1) to x0 and rr (z1) to x1, respectivey, and reduce Eq. 6 to a confuent form of Heun s equation 7,8 Gx x x1 Gx xh xx1 Gx0, 7 with 12iW, 12iW, 4ik, (1iA)iD/k, h2ikia4kw 2Dr. This confuent Heun equation 7, with h its accessory parameter, has two reguar singuar points at x0, 1 with exponents (0, 1) and (0, 1), respectivey, as we as an irreguar singuarity at the infinity point. The power series soution in the vicinity of the point x0 for Eq. 7 can be written as G,,,,h;x g n x n, n0 8 and the coefficient g n satisfies a three-term recurrence reation 7,8 g 0 1, g 1 h/, n1ng n1 n1g n1 nn1hg n. 9 It is not difficut to deduce the exponent 1 soution 8 for x0 and obtain the power series soution in the vicinity of the point x1 by a inear transformation interchanging the reguar singuar points x0 and x1: x 1x. Expansion of soutions to the confuent Heun s equation in terms of hypergeometric and confuent hypergeometric functions has been presented in Refs. 2 and 7. The confuent Heun s functions can be normaized to constitute a group of orthogona compete functions. 7 It shoud be noted that Heun s confuent equation aso admits quasipoynomia soutions for particuar vaues of the parameters. 7,8 It foows from the three-term recurrence reation that G(,,,,h;x) is a poynomia soution if N, with integer N0, g N1 h0, 10 where g N1 being a poynomia of degree N1 inh, that is, there are N1 eigenvaues h i for h such that g N1 (h i )0. IV. HAWKING RADIATION OF SCALAR PARTICLES Now we investigate the Hawking evaporation 9 of scaar partices in the Kerr Sen back hoe by using the Damour Ruffini Sannan s DRS method. 10 This approach ony requires the exis-

4 J. Math. Phys., Vo. 44, No. 3, March 2003 Massive compex scaar fied in Kerr Sen geometry 1087 tence of a future horizon and is competey independent of any dynamica detais of the process eading to the formation of this horizon. The DRS method assumes anayticity properties of the wave function in the compexified manifod. In the foowing, we sha consider a wave going from the event horizon r over interva r r. According to the DRS method, a correct going wave (t,r,,) isan adequate superposition of functions rr and rr, Crr rr r r rr e 2W, 11 where is the conventiona unit step function, C is a normaization factor. In fact, components rr and rr have asymptotic behaviors, rr rr t,r,, c 1 rr iw S m,0 ka,e i(mt) r r, 12 rr rr t,r,, c 2 rr iw S m,0 ka,e i(mt) r r 13 when r r. Ceary, the going wave rr cannot be directy extended from r r to r rr, but it can be anayticay continued to an going wave rr that inside event horizon r by the ower haf compex r-pane around unit circe rr i0: By this anaytica treatment, we have rr r re i. rr c 2 rr iw S m,0 ka,e i(mt). 14 Equation 13 just takes one soution to the radia equation inside the event horizon r, it has the same form of Eq. 14 generated by the anaytica method. As rr differs rr by a factor (rr ) 2iW, then a difference factor e 2W emerges due to the above anaytica treatment. Thus we can derive the reative scattering probabiity of the scaar wave at the event horizon 2 rr e 4W, 15 rr and obtain the therma radiation spectrum with the Hawking temperature T/2. 1 NC 2 e 4W 1, W Ar ma mq, where the anguar veocity at the horizon is a/2mr, the eectric potentia is Q/2M b/q, the surface gravity at the poe is (r Mb)/2Mr /2Mr. The back body radiation spectrum 16 demonstrates that the therma property of Kerr Sen back hoe is simiar to that of Kerr Newman back hoe though its geometry character is ike that of the Kerr soution. 6 Correspondingy, there exist four thermodynamica aws of the Kerr Sen back hoe, simiar to those of Kerr Newman back hoe thermodynamics.

5 1088 J. Math. Phys., Vo. 44, No. 3, March 2003 S. Q. Wu and X. Cai V. CONCLUSION In this paper, we have shown that the separation of variabes of the scaar wave equation in the Kerr Newman back hoe background can appy competey to the case of the twisted Kerr soution. The separated radia part can be recast into the generaized spheroida wave equation, which is, in fact, a confuent form of Heun equation. In addition, we find that the therma property of the twisted Kerr back hoe resembes that of Kerr Newman back hoe though its geometry character ikes that of the Kerr soution. The Kerr Sen soution shares simiar four back hoe thermodynamica aws and quantum therma effect as the Kerr Newman space time does. ACKNOWLEDGMENTS One of us S.Q.W. is very indebted to Dr. Jeff Zhao and Dr. C. B. Yang for their hep in finding some usefu references. 1 S. Q. Wu and X. Cai, J. Math. Phys. 40, E. W. Leaver, J. Math. Phys. 27, ; J.W.Liu,ibid. 33, ; B. D. B. Figneiredo and M. Noveo, ibid. 34, D. R. Bri, P. L. Chrzanowski, C. M. Pereira, E. D. Fackere, and J. R. Isper, Phys. Rev. D 5, ; D.J.Rowan and G. Stephensen, J. Phys. A 10, ; S. Detweier, Phys. Rev. D 22, H. Suzuki, E. Takasugi, and H. Umetsu, Prog. Theor. Phys. 100, ; 102, A. Sen, Phys. Rev. Lett. 69, ; A. Garcia, D. Gatsov, and O. Kechkin, ibid. 74, T. Okai, Prog. Theor. Phys. 92, Henn s Differentia Equations, edited by A. Ronveaux Oxford Science Pubications, Oxford, J. Bandin, R. Pons, and G. Marcihacy, Lett. Nuovo Cimento 38, S. W. Hawking, Nature London 248, ; Commun. Math. Phys. 43, T. Damour and R. Ruffini, Phys. Rev. D 14, ; S. Sannan, Gen. Reativ. Gravit. 20,