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研究生： 何飛宏
Fei-Hung Ho
論文名稱： Teleparallel 理論中之準局域質心距
QUASILOCAL CENTER-OF-MASS FOR GR{II}
指導教授： 聶斯特
James M. Nester
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 物理學系
Department of Physics
畢業學年度： 91
語文別： 英文
論文頁數： 49
中文關鍵詞： asdf
外文關鍵詞： sadf
相關次數： 點閱：16  下載：0 
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    • Nester-Chen 準局域表示式在 Teleparallel 理論及廣義相對論中，可以使能量、動量、角動量及質心距的準局域化 (quasilocalization) 成為協變的 (covariant) ，而此篇論文要討論的是：Teleparallel理論中的準局域質心距，在Nester-Chen 準局域表示式裡佔有重要地位。

    Asymptotically flat gravitating system have 10 conserved quantities associated with
    Poincar´e symmetry, which lack proper local densities. It has been hoped that the
    tetrad formulation and the related teleparallel equivalent of Einstein’s GR (TEGR,
    aka GR{II}) could solve this longstanding gravitational energy-momentum localization
    problem [23, 32, 33]. Quasilocal expressions are now favored. Earlier quasilocal GR{II}
    investigations focused on energy-momentum [32, 33]. Recently our group considered
    angular momentum and found that the popular expression (unlike our “covariantsymplectic”
    one [5]) was not asymptotically locally Lorentz frame gauge invariant;
    it gives the correct result but only in a certain frame [30]. The remaining Poincar´e
    quantity, the center-of-mass moment, has been neglected. Obtaining the correct value
    for this quantity is a quite severe requirement, hence a new discriminating test for
    proposed expressions. We found (independent of the frame gauge choice) that the
    GR{II} “covariant-symplectic” Hamiltonian-boundary-term quasilocal expression succeeds
    while the usual expression does not give the desired center-of-mass moment.
    None of the tetrad expressions gives the desired center-of-mass moment. We conclude
    that the teleparallel formulation is definitely better than the tetrad formulation, and
    the covariant-symplectic expressions are definitely better than the alternatives. We
    also found however that GR{II} has no advantage over GR for energy localization.

    Table of Contents iv Abstract vi Acknowledgements vii 1 Introduction 1 1.1 Conserved Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Symmetry in Physics . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Center-of-Mass Moment . . . . . . . . . . . . . . . . . . . . . 1 1.2 Conserved Quantities for Gravitation . . . . . . . . . . . . . . . . . . 3 1.2.1 Asymptotically Flat . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.2 Pseudotensor for Energy-Momentum . . . . . . . . . . . . . . 3 1.2.3 Quasilocal Quantities . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 GR{II} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Quasilocal Quantities 9 2.1 General Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Noether’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 The Covariant Hamiltonian Approach . . . . . . . . . . . . . . . . . . 12 2.3.1 Hamiltonian Formalism . . . . . . . . . . . . . . . . . . . . . . 12 2.3.2 Covariant Sympletic Quasilocal Expressions . . . . . . . . . . 14 2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3 Teleparallel Formulation 16 3.1 Mathematical Conventions . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 Møller’s Tetrad Representation . . . . . . . . . . . . . . . . . . . . . 17 3.2.1 Lagrange Multiplier Method . . . . . . . . . . . . . . . . . . . 18 3.2.2 Boundary Term . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 General Geometric Theory . . . . . . . . . . . . . . . . . . . . . . . . 21 3.3.1 General Formulation . . . . . . . . . . . . . . . . . . . . . . . 21 3.3.2 General Relativity . . . . . . . . . . . . . . . . . . . . . . . . 22 3.4 Formulation for GR{II} . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 iv 3.4.1 Changing the Variables . . . . . . . . . . . . . . . . . . . . . . 23 3.4.2 GR{II} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.4.3 Quasilocal Quantities for GR{II} . . . . . . . . . . . . . . . . . . 26 4 DN Terms Essential for the Center-of-Mass Moment 29 4.1 Basic Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.1.1 Displacement N¹ . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.1.2 Asymptotically Flat . . . . . . . . . . . . . . . . . . . . . . . 29 4.2 The DN Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2.1 Various B(N) Forms . . . . . . . . . . . . . . . . . . . . . . . 30 4.2.2 Essential DN Terms . . . . . . . . . . . . . . . . . . . . . . . . 31 4.3 Evaluation of The Center-of-Mass Moment . . . . . . . . . . . . . . . 32 4.3.1 Metric and Coframe . . . . . . . . . . . . . . . . . . . . . . . 32 4.3.2 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5 Conclusion 38 Bibliography 40

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