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研究生:黃騰銳
研究生(外文):Teng-Rui Huang
論文名稱:利用漸近疊代方法研究黑洞準正則模的性質
論文名稱(外文):The Asymptotic Iteration Method (AIM) Applied to QNMs of Black Holes
指導教授:曹慶堂
指導教授(外文):Hing-Tong Cho
口試委員:曹慶堂何俊麟陳江梅
口試日期:2011-06-30
學位類別:碩士
校院名稱:淡江大學
系所名稱:物理學系碩士班
學門:自然科學學門
學類:物理學類
論文種類:學術論文
論文出版年:2011
畢業學年度:99
語文別:英文
論文頁數:53
中文關鍵詞:黑洞準正則模漸近疊代方法微擾方程
外文關鍵詞:Black HoleQNMAIMPerturbation
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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。

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.

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 [10] and the WKB method[11]. 20
Table 2. First 10, 20th, 30th Schwarzschild gravitational quasinormal frequencies to four decimal place for l=3 compared with the continued fraction method [10] and the WKB method [11]. 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 [10]. 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 [10]. 49

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