博碩士論文 962402004 詳細資訊




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姓名 劉建良(Jian-Liang Liu)  查詢紙本館藏   畢業系所 物理學系
論文名稱 廣義相對論中以四維度規適配為參考的準局域能量
(4D-metric matching for the reference of quasi-local energy in general relativity)
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摘要(中) 哈密頓三形式扮演了沿著N向量演化方程的生成子的角色。它決定了哈密頓邊界表示 式,也因而決定了準局域量。能量其意義實為能差,能差的概念總是涉及一個相對的參考值,因此無法唯一定義物理的能量。協變哈密頓法[PRD 72 (2005)
104020]指定了一個適當的邊界表示式,而近期的工作中[PRD 84 (2011) 084047;GRG 44 (2011) 2401],考慮球對稱時空的情形,我們藉由四維度規在封閉二維面上的適配條件得到令人滿意的結果。本文分析了一般情形的四維度規在封閉二維面的適配條件。我們發現對於一個二維面,滿足等距嵌入到閔氏空間,在度規適配的條件下仍然具有兩個自由度可以決定參考系的選擇。準局域能量的值形成一個集,若 它是這兩個自由函數的泛函,則臨界點為其一階變分的解,而準局域能量則為相應的臨界值。
摘要(英) The Hamiltonian 3-form plays the role of the generator of the evolution w.r.t. the displacement vector. It is uniquely defined up to a total differential term, the Hamiltonian boundary expression. The latter determines the quasi-local quantities. The meaningful concept of energy involves the difference of the dynamical values w.r.t. the reference values, so that we do not have a unique definition of the physical energies. For the covariant Hamiltonian approach a suitable boundary expression [PRD 72 (2005) 104020] was identified, and in recent works [PRD 84 (2011) 084047; GRG
44 (2011) 2401] we found satisfactory results obtained from matching the four metrics on a 2-sphere for spherically symmetric spacetimes. Here we analyze the general
4D-metric matching on a closed 2-surface. We find that for a 2-surface which satisfies isometric embedding into Minkowski space there are still two degrees of freedom remaining to determine the choice of reference. The quasi-local energy values form a set, and, if it is a functional of the two free functions, the critical values could be determined by the solution of its variation.
關鍵字(中) ★ 四維度規適配
★ 準局域能量
★ 哈密頓量
★ 邊界表示式
★ 等距嵌入
★ 臨界值
關鍵字(英) ★ 4D-metric matching
★ quasi-local energy
★ Hamiltonian
★ boundary expression
★ isometric embedding
★ critical value
論文目次 1 Introduction 1
2 Covariant Hamiltonian approach 4
2.1 First-order Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Hamiltonian formulation . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.2 The boundary conditions . . . . . . . . . . . . . . . . . . . . . 9
2.2.3 Modification of the boundary expression . . . . . . . . . . . . 10
2.2.4 Remark on the conserved quantities . . . . . . . . . . . . . . . 12
2.3 Application to general relativity . . . . . . . . . . . . . . . . . . . . . 13
3 Brief review of our previous works 17
3.1 The choice of reference . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Quasi-local energy for the strategy (i) . . . . . . . . . . . . . . . . . . 19
3.3 Quasi-local energy for the strategy (ii) . . . . . . . . . . . . . . . . . 21
3.3.1 Program I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3.2 Program II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3.3 Alternative approach . . . . . . . . . . . . . . . . . . . . . . . 25
4 Metric matching on a subspace 27
4.1 Notation settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.1.1 Dynamical spacetime (M, g) and a 2-surface S . . . . . . . . . 27
4.1.2 Reference space ( R
1,3 , ¯g) . . . . . . . . . . . . . . . . . . . . . 27
4.1.3 Covariant derivative . . . . . . . . . . . . . . . . . . . . . . . 28
4.1.4 Extrinsic curvature of S . . . . . . . . . . . . . . . . . . . . . 31
i4.2 Matching the four spacetime metric on the closed space-like two surface 31
4.3 The existence of the metric matching . . . . . . . . . . . . . . . . . . 35
5 Application to quasi-local energy in general relativity 45
5.1 General approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.1.1 The holonomic expression in x µ . . . . . . . . . . . . . . . . . 47
5.1.2 The holonomic expression in y a . . . . . . . . . . . . . . . . . 48
5.1.3 The orthonormal frame expression in ϑ
α
. . . . . . . . . . . . 49
5.1.4 The covector basis expression in
¯
ϑ
¯α
. . . . . . . . . . . . . . . 51
5.2 Application to the Kerr like metric . . . . . . . . . . . . . . . . . . . 51
5.2.1 The choice of reference from the variation of the quasi-local
energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2.2 Alternative choice of the controlled variables . . . . . . . . . . 61
6 Conclusion 66
Bibliography 68
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指導教授 聶斯特(James M. Nester) 審核日期 2013-8-28
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