博碩士論文 102222032 詳細資訊




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姓名 林明賢(Ming-sian Lin)  查詢紙本館藏   畢業系所 物理學系
論文名稱
(Characteristics of Cylindrically Symmetric Spacetimes in General Relativity)
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摘要(中) 本論文的主要目的是探討廣義相對論中柱對稱解的特性。首先,我們從柱對稱真空解開始,該解又被稱為列維-奇維塔解;我們嘗試讓列維-奇維塔解的時空座標在弱場效應底下,能夠退回到牛頓力學而使其一致。接下來,我們嘗試著去描述光的重力效應;由於光子不具有對稱性,而難以建立模型描述其重力;因此我們將重點放在一束光的重力效應;然後試著利用該柱對稱模型分別描述並連接光在內部以及外部的時空座標,使其重力效應能得到完整的描述。再來,我們開始尋找能夠隨時間演化的柱對稱解,該期待解能夠同時滿足列維-奇維塔解的形式並隨時間演化。最後,我們開始尋找具有電磁效應的柱對稱解;由於磁場本質上係為柱對稱而非球對稱,因此非常特殊。
摘要(英) The main purpose of this thesis is trying to discuss the characteristics of cylindrically symmetric solutions of general relativity. First, we start from the vacuum solution, which is also called the Levi-Civita solution, and then we are going to make the Levi-Civita solution in weak field to correspond with Newtonian gravity theory with cylindrical symmetry. Next, we will try to use the cylindrical model to describe the gravity effect of light. The gravitational effect for a single photon is diffcult, because we cannot find the symmetry of moving photons easily. So we focus on the gravitational effect for a beam of light, which is cylindrically symmetric. Then, we are try to combine two solutions, corresponding to the interior and exterior, of a beam of light. Afterwards, we try to find a kind of time-evolution solution. The solution not only maintains the Levi-Civita solution form, but also includes time evolution. Finally, we are try to find electric and magnetic solution in cylindrical symmetry. Magnetism is very especial, because it is not spherically symmetric, instead it has cylindrical symmetry.
關鍵字(中) ★ 相對論
★ 柱對稱
★ 光的重力
★ 列維-奇維塔解
關鍵字(英) ★ General Relativity
★ Cylindrically Symmetric
★ Gravitation of Light
★ Levi-Civita Solution
論文目次 1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Research methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Previous Studies 4
2.1 Newtonian Gravity Theory . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Potential Energy Formula . . . . . . . . . . . . . . . . . . . . 4
2.1.2 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Levi-Civita Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Gott’s Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.1 Einstein Equation for Perfect Fluid . . . . . . . . . . . . . . . 8
2.3.2 Gott’s Energy-momentum Assumption . . . . . . . . . . . . . 10
3 Static Cylindrical Solution 12
3.1 Newtonian Approximation . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Geodesic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.4 Weak Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.5 Applications: Gravitational Effects of Light . . . . . . . . . . . . . . 17
3.5.1 Exterior of Light . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.5.2 Interior of Light . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.5.3 Association of Exterior and Interior . . . . . . . . . . . . . . . 18
4 Levi-Civita Solution 20
4.1 Analysis of Potential Energy . . . . . . . . . . . . . . . . . . . . . . . 20
4.1.1 Approximation of Weak Field . . . . . . . . . . . . . . . . . . 20
4.1.2 Difference with Schwarzschild . . . . . . . . . . . . . . . . . . 21
4.2 Numerical Results of Potential Energy . . . . . . . . . . . . . . . . . 22
4.2.1 Energy Equation of a Particle . . . . . . . . . . . . . . . . . . 23
4.2.2 Energy Equation of Light . . . . . . . . . . . . . . . . . . . . 27
4.3 Analysis of Geodesic . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.3.1 Approximation of Weak Field . . . . . . . . . . . . . . . . . . 28
4.3.2 Similarity with Schwarzschild . . . . . . . . . . . . . . . . . . 29
4.4 Numerical Results of Geodesic . . . . . . . . . . . . . . . . . . . . . . 29
4.4.1 Zero Initial Velocity . . . . . . . . . . . . . . . . . . . . . . . 30
4.4.2 Nonzero Initial Velocity . . . . . . . . . . . . . . . . . . . . . 31
5 Time-Evolution of Cylindrical Solutions 35
5.1 Time-Evolution Cylindrical Metric . . . . . . . . . . . . . . . . . . . 36
5.2 Relation with the Kasner Metric . . . . . . . . . . . . . . . . . . . . . 37
5.3 Time-Evolution Levi-Civita Solution . . . . . . . . . . . . . . . . . . 38
5.4 Difference with Schwarzschild . . . . . . . . . . . . . . . . . . . . . . 40
5.5 Time-Evolution Gott’s Solution . . . . . . . . . . . . . . . . . . . . . 41
5.5.1 Consolidation Method . . . . . . . . . . . . . . . . . . . . . . 42
5.5.2 Derived from Kasner Metric . . . . . . . . . . . . . . . . . . . 43
6 Electromagnetic Cylindrical Solutions 45
6.1 Cylindrical Solution with Charge Density . . . . . . . . . . . . . . . . 45
6.2 Cylindrical Solution with Current . . . . . . . . . . . . . . . . . . . . 49
6.3 Electromagnetic Duality . . . . . . . . . . . . . . . . . . . . . . . . . 53
7 Conclusion 55
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[6] H. Weyl, Ann. Phys. Lpz. 54, 117 (1917).
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Wesley, 2003).
[8] J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1999).
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57[12] T. Levi-Civita, Atti Acc. Lincei Rend. 28, 101 (1919).
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(1985).
指導教授 陳江梅(Chiang-Mei Chen) 審核日期 2016-7-4
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