# 臺灣博碩士論文加值系統

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 The general solutions to Klein-Gordon equation in Reissner-Nordstrom black holes are confluent Heun function. However, in order to compute pair production in such system, turning the solution into hypergeometric function is more convenient and necessary due to our good understanding of hypergeometric function and lack of understanding of Heun function. In mathematics and physics literature, there are studies about Heun-to-hypergeometric reduction. In this thesis, we analyze the simplest reduction applied to pair production in RN background.
 Contents1 Introduction 12 Scalar Field in RN Black Holes 32.1 The Reissner-Nordstrom Solution . . . . . . . . 32.2 Probe Charged Scalar Field . . . . . . . . . . . 42.3 Confluent Heun-to-Hypergeometric Reduction . . . 52.4 Conditions for Klein Gordon Equation in RN Background . 63 Spontaneous Pair Production 73.1 Hawking Radiation and Schwinger Effect . . . . . 83.2 Fluxes and Boundary Conditions . . . . . . . . . .93.3 Non-extremal case . . . . . . . . . . . . . . . . 113.4 Extremal case . . . . . . . . . . . . . . . . . . 154 Conclusion 17A SPECIAL EQUATIONS 18A.1 Hypergeometric Function . . . . . . . . . . . . 18A.2 Whittaker Funtion . . . . . . . . . . . . . . . 18A.3 Some Properties of Gamma Functions . . . . . . . 19
 [1] Robert S.Maier,On reducing the Heun equation to the hypergeometric equation, Journal of Dierential Equations Volume 213, Issue 1, 1 June 2005, Pages 171-203[2] Plamen P Fiziev,Exact solutions of Regge Wheeler equation and quasi-normal modes of compact objects, Class. Quantum Grav. 23 2447(2006)[3] P Aydiner, T Birkandan,Physical Problems Admitting Heun-to-Hypergeometric Reduction,arXiv:1401.0449(2015)[4] D. N. Page, hep-th/0612193.[5] P P Fiziev, The Heun functions as a modern powerful tool for research in differentscientific domains, math-ph/1512.04025.[6] P. Aydiner, T. Birkandan, Physical Problems Admitting Heun-to-Hypergeometric Reduction, math-ph/1401.0449[7] J. S. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82, 664 (1951).[8] S. Hawking,"Particles Creation by Black Holes", Commun. Math. Phy.43(1975)[9] S. W. Hawking, Phys. Rev. Lett. 26, 1344 (1971).[10] Larry Smarr, Mass Formula for Kerr Black Holes, Phys. Rev. Lett. 30, 71 Published 8 January 1973; Erratum Phys. Rev. Lett. 30, 521 (1973)[11] J. D. Bekenstein, Ph.D. thesis, Princeton University (1972).[12] C. M. Chen, S. P. Kim, I.C. Lin, J. R. Sun, M. F. Wu, "Spontaneous Pair Production in Reissner-Nordstrom Black Holes", Phys.Rev. D85 (2012) 124041 [arXiv:1202.3224 [hep-th]]
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