準正則模之物理理解

The understanding of quasinormal modes

吳孟勳(Meng-Syun Wu)

指導教授 ： 賀培銘

共同指導教授 ： 高賢忠(Hsien-Chung Kao)

國立臺灣大學/理學院/物理研究所/碩士(2006年)

https://doi.org/10.6342/NTU.2006.01192

摘要

在這篇論文當中，準正則模的物理理解將被討論。傳統上，我們可以藉由考慮一個在史瓦茲度規的Klein-Gordon方程式，在拉普拉斯轉換之下，我們可以從方程式的Wronskian零點來獲得準正則模。但這樣的方式是比較不物理的。我們試圖以散射過程來理解準正則模。在散射的過程裡，我們可以定義S-矩陣，而我們發現那些使S-矩陣的行列式發散的模，與在拉普拉斯轉換下，Wronskian零點定義下的模是等價的。之後我們又使用散射態方法，去建構一個描述散射過程的波函數。我們發現我們可以藉由解波方程式的初始值問題，而去得到準正則模的展開。波函數是物理上可觀測的量，這麼做將使準正則模更為物理。最後我們使用波恩近式去得到半古典的準正則模。在我們的討論底下，準正則模並不只發生在黑洞的情形下，而是一個更普遍的狀況，我們可以用散射問題的角度去理解它。

關鍵字

準正則模 ； 史互茲黑洞 ； 散射矩陣 ； 波恩近似

並列摘要

In this thesis, the physical origin of quasinormal modes is discussed. Traditionally, we can consider a wave equation (ex. Klein-Gordon equation) in the Schwarzschild background, and use the Wronskian of the two independent solutions to the equation in the Laplace picture to define the quasi-normal modes. However, this kind of definition seems to be less physical sense. Therefore we are attempted to use the scattering matrix or S-matrix to understand quasinormal modes. We find the poles of determinant of the S-matrix are equivalent to the zeros of the Wronskian in the Laplace picture. And then we use the scattering states to construct our wave function. We find we can just solve the wave equations with the proper initial conditions in the scattering problem, and then we can use the poles of the determinant of S-matrix to obtain the quasi-normal mode naturally. The wave function is physical, so definition S-matrix is of a more physical sense. Finally, we use Born approximation to obtain the large quasinormal modes in the case of potential barrier and Schwarzschild black hole. We get the not bad results in these cases.

並列關鍵字

quasinormal modes ； Schwarzschild black hole ； scattering matrix ； Born approximation

參考文獻

1 Condon, R.W. and Curney, E.U. 1929 “Quantum mechanics and radioactive disintegration” Phys. Rev. 33 127

2 Corichi, A 2003 “ On quasinormal modes, black hole entropy, and quantum geometry.”Phys. Rev. D67 087502, [gr-qc:/0212126]

3 Gamow, G. 1928 Z. Phys. 51 204

4 Griffiths, D 1994 “Introduction to Quantum Mechanics”, Preutice-Hall 66-68

5 Horowitz, G.T., Hubeny, V.E. 2000 “Quasinormal modes of AdS black holes and the approach to thermal equilibrium” Phys. Rev. D62 024027, [hep-th:/9909056]