單值性近似帶電黑洞的粒子對產生與準正規模式;Monodromy Approach to Pair Production and Quasi-Normal Modes of Charged Black Holes

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題名:	單值性近似帶電黑洞的粒子對產生與準正規模式;Monodromy Approach to Pair Production and Quasi-Normal Modes of Charged Black Holes
作者:	魏君宇;Wei, Chun-Yu
貢獻者:	物理學系
關鍵詞:	帶電黑洞;單值性;粒子對產生;準正規模式;Charged Black Holes;Monodromy;Pair Production;Quasi-Normal Mode
日期:	2025-07-28
上傳時間:	2025-10-17 11:56:09 (UTC+8)
出版者:	國立中央大學
摘要:	探討在純量場中帶電的黑洞周圍時，四奇點黎曼方程的「匯合形式」 (confluent form) 就會成為重要的研究主題。這是因為，當我們分析「克萊恩-戈登」 (Klein-Gordon) 方程，並且考慮有質量的粒子掉入帶電黑洞的背景下，便會出現四奇點黎曼方程的匯合形式。並且，這類方程屬於「匯合式休恩方程族」 (confluent Heun equations)，因此討論休恩方程也成了本論文的研究主軸。但是，這類方程尚未被完全理解，因為其「附加參數」(accessory parameter) 的解析極為困難。幸運的是，從物理的觀點來看，我們的目標並不是要理解黑洞本身，而是要獲取從黑洞傳出的資訊。因為我們無法進入黑洞內部或靠近其周圍取得資訊，所以我們改為選擇接收來自黑洞所發出的輻射。若用數學語言來說，我們關心的是「穿透係數」 (transmitting coefficient) 的大小，而非解出微分方程本身。面對這個問題，我採用一種有趣的數學技巧，稱為「單值性」（monodromy）方法來處理。透過單值性技術，只需由微分方程的「特徵指數」 (characteristic exponent) 就可以求得穿透係數，因此無需解出方程的精確解。在本論文中，我考慮了兩種邊界條件，來分析黑洞所傳遞出的資訊：其一是「粒子對產生」（pair production），另一種則是「準正規模式」（quasi-normal mode）。最後，為了驗證用單值化方法所得到的結果，我將本論文中的結果與已發表的論文進行比較，並得出結論：本研究所使用的技術是一種可靠的方法。;The confluent form of the four-singularity Riemann equation arises when we investigate phenomena induced by a scalar field in the surroundings of a charged black hole. If we analyze the Klein-Gordon equation with particles incident on (and emitted from) the charged black holes, the corresponding differential equation belongs to the family of confluent Heun equations, which is the main topic of this thesis. So far in the literature, this kind of equation has not been solved completely. Technically, the reasons for these difficulties are related to the accessory parameter $\tilde{q}$. Fortunately, from the physics point of view, we do not need to solve the general scattering (or tunneling) problem of the black hole because only the information we can detect from the black hole is available at very far distances. Therefore, we can use the detected radiation from them, impose suitable boundary conditions, and be able to find the magnitude of the transmitting coefficient, not the solution of the differential equation itself. To handle this task, we use an interesting mathematical technique, called monodromy. Using monodromies, the transmitting coefficient is obtained by the characteristic exponents of the differential equation, so we do not need to know the exact solution. In this thesis, two kinds of boundary conditions are considered to estimate the information of black holes. One corresponds to pair production of particles and the other to quasi-normal modes. To verify the results from monodromy technique applied to Heun-type differential equations, we compare the calculations with the results published in previous papers by the author and her collaborators, as well as with other results available in the literature.
顯示於類別:	[物理研究所] 博碩士論文

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