臺灣博碩士論文加值系統

在帶電的黑洞，萊斯納-諾德斯特洛姆黑洞的時空背景下，克萊恩-戈登方程式的解是合流Heun函數。然而，在這樣的系統下，要計算粒子自發性的成對產生，將解轉換成超幾何函數是更方便，甚至是必要的，由於我們對超幾何函數的有完整理解，但對合流Heun函數上缺乏一個完整的理論。在數學和物理的文獻中，已有將合流Heun函數簡化為超幾何函數的討論及應用。本篇論文將分析，將平凡的合流Heun函數簡化成超幾何函數，用來計算在萊斯納-諾德斯特洛姆黑洞的時空背景下，非極端和極端黑洞的粒子自發性的成對產生。但由於簡化條件對系統參數的限制，使得我們仍無法定量地討論帶電黑洞自發性的成對產生的兩個機制，Hawking radiation和Schwinger pair production。

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.

Contents
1 Introduction 1
2 Scalar Field in RN Black Holes 3
2.1 The Reissner-Nordstrom Solution . . . . . . . . 3
2.2 Probe Charged Scalar Field . . . . . . . . . . . 4
2.3 Confluent Heun-to-Hypergeometric Reduction . . . 5
2.4 Conditions for Klein Gordon Equation in RN Background . 6
3 Spontaneous Pair Production 7
3.1 Hawking Radiation and Schwinger Effect . . . . . 8
3.2 Fluxes and Boundary Conditions . . . . . . . . . .9
3.3 Non-extremal case . . . . . . . . . . . . . . . . 11
3.4 Extremal case . . . . . . . . . . . . . . . . . . 15
4 Conclusion 17
A SPECIAL EQUATIONS 18
A.1 Hypergeometric Function . . . . . . . . . . . . 18
A.2 Whittaker Funtion . . . . . . . . . . . . . . . 18
A.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 different
scientific 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-Nordstrom Black Holes", Phys.Rev. D85 (2012) 124041 [arXiv:1202.3224 [hep-th]]