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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