廣義相對論中以四維度規適配為參考的準局域能量

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研究生：	劉建良 Jian-Liang Liu
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
指導教授：	聶斯特 James M. Nester
口試委員:
學位類別：	博士 Doctor
系所名稱：	理學院 - 物理學系 Department of Physics
論文出版年：	2013
畢業學年度：	101
語文別：	英文
論文頁數：	81
中文關鍵詞：	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
外文關鍵詞：	4D-metric matching, quasi-local energy, Hamiltonian, boundary expression, isometric embedding, critical value
相關次數：	點閱：12 下載：0
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查詢本校圖書館目錄查詢臺灣博碩士論文知識加值系統勘誤回報

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。

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.

Introduction 1
Covariant Hamiltonian approach 4
1 First-order Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Hamiltonian formulation . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 The boundary conditions . . . . . . . . . . . . . . . . . . . . . 9
2.3 Modiﬁcation of the boundary expression . . . . . . . . . . . . 10
2.4 Remark on the conserved quantities . . . . . . . . . . . . . . . 12
3 Application to general relativity . . . . . . . . . . . . . . . . . . . . . 13
Brief review of our previous works 17
1 The choice of reference . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Quasi-local energy for the strategy (i) . . . . . . . . . . . . . . . . . . 19
3 Quasi-local energy for the strategy (ii) . . . . . . . . . . . . . . . . . 21
3.1 Program I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Program II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Alternative approach . . . . . . . . . . . . . . . . . . . . . . . 25
Metric matching on a subspace 27
1 Notation settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.1 Dynamical spacetime (M, g) and a 2-surface S . . . . . . . . . 27
1.2 Reference space ( R
1,3 , ¯g) . . . . . . . . . . . . . . . . . . . . . 27
1.3 Covariant derivative . . . . . . . . . . . . . . . . . . . . . . . 28
1.4 Extrinsic curvature of S . . . . . . . . . . . . . . . . . . . . . 31
i4.2 Matching the four spacetime metric on the closed space-like two surface 31
3 The existence of the metric matching . . . . . . . . . . . . . . . . . . 35
Application to quasi-local energy in general relativity 45
1 General approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
1.1 The holonomic expression in x µ . . . . . . . . . . . . . . . . . 47
1.2 The holonomic expression in y a . . . . . . . . . . . . . . . . . 48
1.3 The orthonormal frame expression in ϑ
α
. . . . . . . . . . . . 49
1.4 The covector basis expression in
¯
ϑ
¯α
. . . . . . . . . . . . . . . 51
2 Application to the Kerr like metric . . . . . . . . . . . . . . . . . . . 51
2.1 The choice of reference from the variation of the quasi-local
energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.2 Alternative choice of the controlled variables . . . . . . . . . . 61
Conclusion 66
Bibliography 68
                                

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