Teleparallel 理論中之準局域質心距

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姓名

何飛宏(Fei-Hung Ho) 查詢紙本館藏

畢業系所

物理學系

論文名稱

Teleparallel 理論中之準局域質心距
(QUASILOCAL CENTER-OF-MASS FOR GR{II})

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摘要(中)

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。

摘要(英)

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.

關鍵字(中)

★ asdf

關鍵字(英)

★ sadf

論文目次

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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指導教授

聶斯特(James M. Nester)

審核日期

2003-7-24

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