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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 [1] J. D. Jackson, Classical Electrodynamics.
[2] S. Chandrasekhar, The Mathematical Theory of Black Holes .
[3] J. B. Griffiths and J. Podolsky, Exact Space-Times in Einstein’s General
Relativity.
[4] Da-Ping Du and Bin Wang, Phys.Rev.D.70.064024
[5] Avis S J, Isham C J and Storey D, 1978 Phys. Rev. D 18 3565
[6] Akihiro Ishibashi and Robert MWald, Class. Quantum Grav. 21 (2004)
2981–3013
[7] C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists
and Engineers.
[8] H. T. Cho and Y. C. Lin, 2005 Class. Quantum Grav. 22 775
[9] H. P. Nollert, Classical Quantum Gravity 16, R159 (1999)
[10] 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.
[11] Sai Iyer, Phys. Rev. D 35, 3632–3636 (1987).
[12] A. Zhidenko, Class. Quantum Grav. 21 273 (2004 ) (arXiv:gr-qc/0307012)
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