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研究生：	陳威辰 Chen, Wei-Chen
論文名稱：	任意彎曲時空中相對論性時間效應的動力學準則 Dynamical basis of relativistic time effects in generic curved spacetimes
指導教授：	許祖斌 Soo, Chopin
學位類別：	碩士 Master
系所名稱：	理學院 - 物理學系 Department of Physics
論文出版年：	2018
畢業學年度：	106
語文別：	英文
論文頁數：	45
中文關鍵詞：	相對論性時間效應、逝函數、折合哈密頓量、宇宙時間
外文關鍵詞：	relativistic time effects, emergent lapse function, reduced Hamiltonian, cosmic time
相關次數：	點閱：140 下載：15
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本論文探討了任意時空下相對論性時間效應的動力學準則。首先推導出測試粒子於任意彎曲時空以 Arnowitt-Deser-Misner 形式表現的哈密頓量，進而用以探討相對論性的時間效應。所得到的方程式囊括了狹義相對論中相對運動及廣義相對論中不同重力為勢所造成的影響，並可應用於全球定位系統及重力紅移的原子物理實驗。藉由連結廣義相對論中的逝函數與內稟宇宙時間演化論的折合哈密頓量，相對論性的時間效應可以歸因為能量的差異。

The dynamical basis of relativistic time effects in generic curved spacetimes is investigated.
The Hamiltonian of a test particle in a generic Arnowitt-Deser-Misner spacetime is derived, and employed to discuss relativistic time effects.
This generalizes the usual special and general relativistic effects, due respectively to relative motion and difference in gravitational potential, into a single formula which, among other benefits, is directly applicable to global positioning systems and atomic experiments of gravitational redshift.
By relating the a posteriori emergent lapse function in general relativity to the reduced Hamiltonian for intrinsic cosmic time evolution, all the relativistic time effects discussed can trace their dynamical origins to differences in energy.

摘要	i
Abstract	ii
誌謝	iii
Contents	iv
List of Figures	vi
Numerical values of some physical quantities	vii
Chapter 1  Introduction and Overview	1
Chapter 2  Test Particle in Gravitational Field	3
2.1  Introduction   ............      3
2.2  Arnowitt-Deser-Misner decomposition  ........      3
2.3  Lagrangian and Hamiltonian ..........      4
2.3.1  Action of a test particle   ........      4
2.3.2  Lagrangian and conjugate momentum .......      5
2.3.3  Hamiltonian and Hamilton’s equations  ......      5
2.4  Time dilation effects ..........      6
2.4.1  Proper time ...........      6
2.4.2  Examples of time dilation effects  .......      7
2.5  Lorentz boost and its relation to conjugate momentum of particle  ..      9
2.5.1  Vierbein   ............      9
2.5.2  4-momentum ..........    10
2.5.3  Explicit Lorentz boost  .........    13
Chapter 3  Applications	14
3.1  Introduction   ............    14
3.2  Global Positioning System (GPS)  .........    14
3.2.1  Current correction  .........    14
3.2.2  Special relativistic effect ........    14
3.2.3  General relativistic effect   .........    15
3.2.4  Total effect by naive addition  .......    15
3.2.5  Exact correction  ..........    16
3.2.6  Calculation with Painlevé-Gullstrand (PG) metric   ....    17
3.3  Frequency shift of signals  .........    18
3.3.1  Solutions of Maxwell’s equations  .......    18
3.3.2  Redshift frequency .........    19
3.3.3  Atomic experiments of gravitational redshift  .....    20
Chapter 4  Hamiltonian Formulation of General Relativity	21
4.1  Introduction   ............    21
4.2  Decomposition of Ricci scalar .........    21
4.2.1  Normal vector  ...........    21
4.2.2  3-metric and projector  .........    22
4.2.3  3-covariant derivative  .........    23
4.2.4  Riemann-Christoffel tensor  ........    24
4.2.5  Extrinsic curvature .........    25
4.2.6  Ricci Scalar   ..........    26
4.3  Lagrangian and Hamiltonian ..........    29
4.3.1  Einstein-Hilbert action .........    29
4.3.2  Lagrangian and conjugate momenta ......    29
4.3.3  Hamiltonian and constraints ........    31
4.4  Symplectic potential decomposition ........    32
4.4.1  Introduction  ..........    32
4.4.2  Lapse function ...........    34
Chapter 5  Concluding Remarks	36
References	37
Appendix A  Painlevé-Gullstrand Metric	38
A.1  Introduction  ............    38
A.2  Coordinate singularity in Schwarzschild metric   .....    38
A.3  Painlevé-Gullstrand metric  ..........    38
Appendix B  Lie Derivative and Killing Vectors	42
B.1  Lie derivative  ...........    42
B.2  Killing vector  ...........    43
Appendix C  Hamiltonian of Test Particle	44
C.1  Introduction  ............    44
C.2  Motion of test particle ...........    44
                                    

[1] C. Soo and H.-L. Yu, Prog. Theor. Exp. Phys. 2014, 013E01 (2014).
[2] R. Arnowitt, S. Deser, and C. W. Misner, Phys. Rev. 116, 1322 (1959).
[3] C. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Princeton University Press, New
Jersey, 2017).
[4] B. W. Parkinson and J. J. S. Jr., Global Positioning System: Theory and Applications, Volume I
(American Institue of Aeronautics and Astronautics, Washington, 1996).
[5] R. M. Wald, General Relativity (University of Chicago Press, Chicago, 1984).
[6] M. A. Hohensee, B. Estey, F. Monsalve, G. Kim, P.-C. Kuan, S.-Y. Lan, and H. Mueller, J. Phys.
Conf. Ser. 264, 012009 (2011).
[7] C. Soo and H.-L. Yu, Chin. J. Phys. 53, 110102 (2015).
[8] C. Soo, Int. J. Mod. Phys. D 25, 1645008 (2016).
[9] E. E. I. III, C. Soo, and H.-L. Yu, Prog. Theor. Exp. Phys. 2015, 083E01 (2015).
[10] H.-C. Lin and C. Soo, Chin. J. Phys. 53, 110106 (2015).

校內：2020-06-15公開
校外：2020-06-15公開

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