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 系統識別號 U0002-2607201117282400 中文論文名稱 利用漸近疊代方法研究黑洞準正則模的性質 英文論文名稱 The Asymptotic Iteration Method (AIM) Applied to QNMs of Black Holes 校院名稱 淡江大學 系所名稱(中) 物理學系碩士班 系所名稱(英) Department of Physics 學年度 99 學期 2 出版年 100 研究生中文姓名 黃騰銳 研究生英文姓名 Teng-Rui Huang 學號 697210135 學位類別 碩士 語文別 英文 口試日期 2011-06-30 論文頁數 53頁 口試委員 指導教授-曹慶堂委員-曹慶堂委員-何俊麟委員-陳江梅 中文關鍵字 黑洞  準正則模  漸近疊代方法  微擾方程 英文關鍵字 Black Hole  QNM  AIM  Perturbation 學科別分類 學科別＞自然科學＞物理 中文摘要 我們在這篇論文裡說明如何運用漸近疊代方法(the asymptotic iteration method)，計算四維時空裡不同黑洞(Schwarzschild、Reissner-Nordström和Kerr)的準正則模(quasinormal modes)。對於Schwarzschild黑洞，我們計算重力微擾的準正則頻率。至於Kerr黑洞，我們則計算純量和重力微擾的準正則頻率。我們特別討論低模的數值結果，並且和之前發表的結果做比較。 英文摘要 In this thesis we show how to use the asymptotic iteration method (AIM) to numerically calculate the quasinormal modes (QNMs) of different (Schwarzschild, Reissner-Nordström and Kerr) black holes in four-dimensional spacetime. For Schwarzschild black holes, we compute the quasinormal frequencies of the gravitational perturbations. For the Kerr black holes, we consider both the scalar and the gravitational cases. We discuss our results especially for the low-lying modes, and compare them to previously published results. 論文目次 Contents Chapter 1 Introduction 1 1.1 The QNMs of black holes 1 1.2 Formalism of the AIM 2 Chapter 2 Schwarzschild black holes 7 2.1 Radial perturbation equation for Schwarzschild black holes 7 2.2 The AIM for determining the quasinormal frequencies of Schwarzschild black holes 11 2.3 The numerical results 16 Chapter 3 Reissner-Nordström black holes 24 3.1 Radial perturbation equations for Reissner-Nordström black holes 24 3.2 The AIM for determining the quasinormal frequencies of Reissner- Nordström black holes 27 3.3 The numerical results 30 Chapter 4 Kerr black holes 34 4.1 Angular and radial perturbation equations for Kerr black holes 34 4.2 The AIM for determining the quasinormal frequencies of Kerr black holes 37 4.3 The numerical results 46 Chapter 5 Conclusions 50 References 52 Figures and Tables Figure 1. Regge-Wheeler and Zerilli potentials for l=2 and l=3 for gravitational perturbation. 11 Figure 2. The Regge-Wheeler potential for l=2 to5. 16 Figure 3. ξ for l=2 to 30. 17 Figure 4. The Schwarzschild gravitational quasinormal frequencies for l=2 by the AIM. 19 Table 1. First 10, 20th, 30th Schwarzschild gravitational quasinormal frequencies to four decimal place for l=2 compared with the continued fraction method  and the WKB method. 20 Table 2. First 10, 20th, 30th Schwarzschild gravitational quasinormal frequencies to four decimal place for l=3 compared with the continued fraction method  and the WKB method . 22 Figure 5. The trend of Schwarzschild gravitational quasinormal frequencies of different n for l=2 from the number of iterations 60 to 300 with step 20. 22 Figure 6. First 5 Schwarzschild gravitational quasinormal frequencies for l=2 to 30. Fundamental mode is at the top, fifth overtone at the bottom. The quasinormal frequencies go from left to right when l is increased. 23 Table 3. Reissner-Nordström quasinormal frequency parameter values for the fundamental and two lowest overtones for l=2 and i=2. 31 Table 4. Reissner-Nordström quasinormal frequency parameter valuesfor the fundamental and two lowest overtones for l=2 and i=1. 32 Table 5. Reissner-Nordström quasinormal frequency parameter valuesfor the fundamentaland two lowest overtones for l=1 and i=1. 33 Figure 7. Kerr scalar quasinormal frequencies for the fundamental and first overtones, for l=1. Values shown for a=0, .1, .2, .3, .4, .45. 47 Figure 8. Kerr scalar quasinormal frequencies for the first 3 overtones, for l=2. Values shown for a=0, .1, .2, .3, .4, .45. 47 Table 6. Angular separation constants and Kerr gravitational quasinormal frequencies for the fundamental mode corresponding to l=2 and m=0 compared with the continued fraction method . 48 Table 7. Angular separation constants and Kerr gravitational quasinormal frequencies for the fundamental mode corresponding to l=2 and m=1 compared with the continued fraction method . 49 參考文獻  H. Ciftci, R. L. Hall and N. Saad, J. Phys. A: Math. Gen. 36, 11807 (2003).  H. T. Cho, A. S. Cornell, J. Doukas, and W. Naylor, Class. Quant. Grav. 27, 155004 (2010).  T. Regge and J. A. Wheeler, Phys. Rev. 108, 1063 (1957).  F. J. Zerilli, Phys. Rev. D 2, 2141 (1970).  S. A. Teukolsky, Phys. Rev. Lett. 29, 1114 (1972).  F. J. Zerilli, Phys. Rev. D 9, 860 (1974).  V. Moncreif, Phys. Rev. D 9, 2707 (1974); 10, 1057 (1974); 12, 1526 (1975).  H. P. Nollert, Class. Quant. Grav. 16, R159 (1999).  E. Berti, V. Cardoso and A. O. Starinets, Class. Quant. Grav. 26, 163001 (2009).  E. W. Leaver, Proc. R. Soc. A 402, 285 (1985).  S. Iyer, Phys. Rev. D 35, 3632 (1987).  S. Chandrasekhar, Proc. R. Soc. Lond. A 392, 1 (1984).  H. P. Nollert, Phys. Rev. D 47, 5253 (1993).  S. Hod, Phys. Rev. Lett. 81, 4293 (1998).  K. D. Kokkotas and B. Schutz, Phys. Rev. D 37, 3378 (1988).  E. W. Leaver, Phys. Rev. D 41, 2986 (1990).  S. Chandrasekhar, The Mathematical Theory of Black Holes, page 237 (Clarendon, Oxford, 1983).  K. D. Kokkotas, Class. Quant. Grav. 8, 2217 (1991).  S. Detweiler, Proc. R. Soc. London A352, 381 (1977).  E. Seidel and S. Iyer, Phys. Rev. D 41, 374 (1990). 論文使用權限 同意紙本無償授權給館內讀者為學術之目的重製使用，於2016-07-28公開。同意授權瀏覽/列印電子全文服務，於2016-07-28起公開。 若您有任何疑問，請與我們聯絡！圖書館： 請來電 (02)2621-5656 轉 2487 或 來信 dss@mail.tku.edu.tw 