          下載電子全文 （限經由淡江IP使用）
 系統識別號 U0002-2106201215392800 中文論文名稱 在球對稱時空下的馬克斯威方程組 英文論文名稱 Maxwell Equations in Spherically Symmetric Spacetimes 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 100 學期 2 出版年 101 研究生中文姓名 蔡瓈篁 研究生英文姓名 Li-Huang Tsai 學號 698210126 學位類別 碩士 語文別 英文 口試日期 2012-06-18 論文頁數 71頁 口試委員 指導教授-曹慶堂委員-劉國欽委員-陳江梅 中文關鍵字 馬克斯威方程組  球對稱  WKB 近似 英文關鍵字 Maxwell equations  spherically symmetric  WKB approximation 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，討論在不同的球對稱時空下的馬克斯威方程組。在極大對稱的時空裡，我們找出精確解；對於其他的球對稱時空，我們求助於WKB 的近似方法去找出馬克斯威方程組的解。並根據其適當的邊界條件，我們得以在不同的問題中得到散射震，束縛能量，以及准正則模的頻率。 英文摘要 In this thesis, we discuss the Maxwell equations in different spherically symmetric spacetimes. In maximally symmetric spacetime, we find exact solutions; for the other spacetimes, we resort to the WKB approximation method to find solutions to the Maxwell equations. We obtain scattering amplitudes, bound state energies, and quasi-normal mode frequencies according to the appropriate boundary conditions in the problem.. 論文目次 Chapter1. Introduction 1 Chapter2. General Formalism of Maxwell Equations in Spherically Symmetric Spacetimes 3 (i) Maxwell Equations in Curved Spacetime 3 (ii) Formalism 5 (a) Polar case 6 (b) Axial case 9 (iii) The Isospectral Property of the Polar and Axial Cases 12 Chapter3. The Exact Solutions of Maxwell Field Equations in Minkowski , de Sitter , and Anti-de Sitter Spacetimes 14 (i) The Exact Solutions in Minkowski Spacetime 15 (ii) The Exact Solutions in deSitter and Anti- deSitter Spacetimes 18 (A) deSitter Spacetime 21 (B) Anti- deSitter Spacetime 24 Chapter4. WKB Approximations of the Scattering Problem for Maxwell Equations in Schwarzschild and Reissner – Nordstrom Spacetimes 29 (i) Spherically Symmetric Black Holes Spacetimes 29 (ii) WKB approximation 30 (A) Simple WKB Cases for One and Two turning points 30 (B) Application 35 (iii) Scattering problems 41 (A) Schwarzschild spacetime 41 (a) Transmission probabilities with l=1 43 (b) Transmission probabilities with variable l 45 (B) Reissner-Nordstrom spacetime 48 (a) Transmission probabilities with l = 1, variable Q 49 (b) Transmission probabilities with different l , variable Q 51 Chapter5. QNM problems for Maxwell Field Equations in some Spherically Symmetric Spacetimes 57 (i) Schwarzschild spacetime 57 (ii) Reissner-Nordstrom spacetime 62 (iii) Schwarzschild de Sitter spacetime 63 Chapter6. Conclusion 69 References 71 Fig.(3.1) : The arc hyperbolic tangent function 3 0.3 tanh-1 ( 0.3r ) 18 Fig.(3.2) : The arc hyperbolic tangent function 3 -0.3 tanh-1 ( -0.3r ) 18 Fig.(3.3) : The effective potential of axial field in de Sitter spacetime 19 Fig.(3.4) : The effective potential of axial field in anti-de Sitter spacetime 19 Fig.(3.5) : The Gamma function G(x) plotted in the range -10 < x < 4 23 Fig.(4.1) : As 0 r R r , V(r) ~ar , a < 0 . V (r) - E = 0 30 Fig.(4.2) : As 0 r R r , V(r) ~ar , a > 0 . V (r) - E = 0 32 Fig.(4.3) : As 0 r R r , V(r) behaves as parabolic with V(r) ~ ar 2 , a < 0 . V (r) - E = 0 . 33 Fig.(4.4) : The shape of a barrier potential for scattering problem 35 Fig.(4.5) : Variation of effective potential with l of Maxwell field in tortoise coordinate * r 42 Fig.(4.6) : Validity condition for various values of w of the Maxwell fields with l =1 43 Fig.(4.7) : Transmission probabilities of the Maxwell field with l =1 in the various WKB approximations for 2 m w ≫V (dotted line), 2 m w »V (solid line), and 2 m w ≪V (dashed line). 44 Fig.(4.8) : Transmission probabilities of the Maxwell field with l =1 44 Fig.(4.9) , (I ) ~ (V ): Validity condition for various values of w of the Maxwell fields with l = 2 ~ 6 45 Fig.(4.10) , (I ) ~ (V ): Transmission probabilities of the Maxwell field in Schwarzschild spacetime with l=2~6 in various WKB approximations for 2 m w ≫V (dotted line), 2 m w »V (solid line), and 2 m w ≪V (dashed line). 46 Fig.(4.11) : Transmission probabilities T of the Maxwell field in Schwarzschild spacetime with l = 2 ~ 6 47 Fig.(4.12) :Effective potential in * r in RN spacetime 49 Fig.(4.13) , (I ) ~ (VI ) : The cross point of transmission probabilities of the Maxwell field in RN spacetime with various WKB approximations for 2 m w » V (solid line), and 2 m w ≪V (dashed line). 49-50 Fig.(4.14) : Transmission probabilities of the Maxwell field in RN spacetime with l=1, Q =0, 0.2, 0.4, 0.6, 0.8, and 1. 51 Fig.(4.15) : Transmission probabilities of the Maxwell field in RN spacetime with l=2, Q =0, 0.2, 0.4, 0.6, 0.8, and 1. 52 Fig.(4.16) :Transmission probabilities of the Maxwell field in RN spacetime with l=3, Q =0, 0.2, 0.4, 0.6, 0.8, and 1. 52 Fig.(4.17), (I ) ~ (IV ) : Transmission probabilities of the Maxwell field in RN spacetime with Fixed Q 53 Fig.(4.18), (I ) : m V is fixed, small interval for l Fig.(4.18), (II ) : m V is fixed, large interval for l Fig.(4.18), (III ) : l is fixed, small interval for m V Fig.(4.18), (IV ) : l is fixed, large interval for m V Fig.(4.18), (V ) : fixed ratio of m V and l 54 Fig.(5.1) : The function of (V(r)-E) 58 Fig.(5.2) : Quasi-normal frequencies with various values of parameters l, n in Schwarzschild spacetime. Blue for n = 0 , red for l =1, and black for l = 2 61 Fig.(5.3) : Quasi-normal frequencies with various values of parameters Q,l,n in RN spacetime. Blue for l =1, red for l = 2, and black for l = 3 . 63 Fig.(5.4) : The metric f in SdS spacetime, with different values of Lɶ 64 Fig.(5.5) : Variation of effective potentials with Lɶ of Maxwell field in r 66 Fig.(5.6) : Quasi-normal frequencies with various values of parameters Lɶ , l, n in SdS spacetime. Blue for l =1 , red for l = 2 , and black for l = 3 67 參考文獻  J. D. Jackson, Classical Electrodynamics.  S. Chandrasekhar, The Mathematical Theory of Black Holes .  J. B. Griffiths and J. Podolsky, Exact Space-Times in Einstein’s General Relativity.  Da-Ping Du and Bin Wang, Phys.Rev.D.70.064024  Avis S J, Isham C J and Storey D, 1978 Phys. Rev. D 18 3565  Akihiro Ishibashi and Robert MWald, Class. Quantum Grav. 21 (2004) 2981–3013  C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers.  H. T. Cho and Y. C. Lin, 2005 Class. Quantum Grav. 22 775  H. P. Nollert, Classical Quantum Gravity 16, R159 (1999)  Schutz, B. F.; Will, C. M., Astrophysical Journal, Part 2 - Letters to the Editor (ISSN 0004-637X), vol. 291, April 15, 1985, p. L33-L36.  Sai Iyer, Phys. Rev. D 35, 3632–3636 (1987).  A. Zhidenko, Class. Quantum Grav. 21 273 (2004 ) (arXiv:gr-qc/0307012) 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2012-06-29公開。同意授權瀏覽/列印電子全文服務，於2012-06-29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 2486 或 來信 