廣義相對論的準局域量的小球極限

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姓名

孫綱(Gang Sun) 查詢紙本館藏

畢業系所

物理學系

論文名稱

廣義相對論的準局域量的小球極限
(Quasilocal Conserved Quantities For General Relativity In Small Regions)

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摘要(中)

引力能量是個長期被關注的問題。到目前為止，有幾種方法可以處理這個問題，但最好的辦法是使用準局域的概念。中央大學幾何物理學研究團隊已經發展出他們的準局域的理論－協變哈彌爾頓架構，並且在類光和類空無窮遠獲得好的計算結果。不過，實際上目前在大尺度的真空中極限僅僅是檢驗了弱場線性引力的準局域表達。除了無窮遠的情況，還存在另一類的極端情形－小區域的極限。真空的小球極限是對準局域的理論的重要測試，這種測試是檢驗引力場第一階的非線性部分。
這篇論文的目的是去測試協變哈彌爾頓架構在小範圍的極限。在第一章裡，我們將介紹準局域的基本概念和相關的應用；在第二章，我們將告訴讀者什麼是協變哈彌爾頓架構並且推導它；在第三章，我們將介紹能量、動量、角動量和質心矩的一般性概念並展示物理量和守恆間的關係；在之後的章節，用準局域的方法計算小球極限的量值過程將會詳細地被說明，此處，我們不但考慮了真空，也給出了含有物質場的情況；最後一章裡，我們會討論計算結果所隱含的意義，而這些結果告訴我們對四種協變哈彌爾頓的表示式僅有一種給出了正能量。

摘要(英)

Gravitational energy has been a concern for a long time. There are several ways to deal with the problem, but the best way is the quasilocal approach. The NCU group has been developing their quasilocal approach – the covariant Hamiltonian formalism, and has obtained good results for spatial and null infinity. In addition to these infinite cases; there is another limit case – the small region limit. The small region vacuum limit provides an important test of the quasilocal expression. Whereas the large scale asymptotic limit tests only the weak field linearized part of the expression, the small scale vacuum limit probes the next order non-linear part.
In this thesis the purpose is to test the covariant Hamiltonian formalism in the small region limit. In the first chapter, we will introduce the basic ideas of the quasilocal method and some related ideas. In chapter two, we will show the readers what the covariant Hamiltonian formulism is and how to derive it. In the chapter three, we will introduce some general concepts about energy-momentum, angular momentum and center-of-mass moment, and the relation between these physical quantities and conservation. In the next chapter, the detailed procedure on how to get quasilocal values in the small region limit, including the vacuum case and matter case, using covariant Hamiltonian formulism will follow. In the final chapter, we will discuss the meaning of our results and conclude that only one of the four covariant Hamiltonian expressions gives positive energy in the first non-linear order. Finally we will comment some deficiencies.

關鍵字(中)

★ 小球極限
★ 準局域

關鍵字(英)

★ small region limit
★ quasilocal

論文目次

Contents
Chapter 1
Introduction 1
1.1 Fundamental concepts of quasilocal 1
1.2 Some applications of quasilocal 2
1.2.1 Tidal heating 2
1.2.2 Positivity of the gravitational energy in the finite region 3
1.2.3 The ADM mass and the (weak) cosmic censorship hypothesis 3
Chapter 2 4
Covariant symplectic quasilocal expression 4
2.1 From Lagrangian to Hamiltonian 4
2.2 Control over the Hamiltonian boundary term 6
2.3 The choice of evolving vector 9
2.3.1 Asymptotically 9
2.3.2 In the small region 12
Chapter 3 14
Physical quantities and conservation 14
3.1 Physical quantities in special relativity 14
3.1.1 Single particle 14
3.1.2 Matter field 15
3.2 Conservation Law and Noether theorem 16
3.2.1 Killing vector field 16
3.2.2 Noether theorem 17
3.2.3 Conservation quantities in Minkowski space 18
3.3 Total conservation for a gravitational field 19
3.3.1 Pseudotensor 19
3.3.2 Our quasilocal approach 20
Chapter 4 21
Quasilocal quantities in small region 21
4.1 Riemann normal coordinates 21
4.2 The expansion of the Hamiltonian in the small region 23
4.3 The quasilocal values expressed by the different unit volume 32
4.4 The values of the four quasilocal cases 36
4.4.1 The four cases expressed by the physical volume element 36
4.4.2 The four cases expressed by the flat volume element 38
Chapter 5 41
Discussion 41
5.1 Discussion to the results including a matter field 41
5.2 Discussion to those results concerning the vacuum 41
5.2.1 The physical quantities in the small region 41
5.2.2 Some comments on our quasilocal values in vacuum 46
5.3 Conclusion 50
Reference 53
Appendix A 55
Bel-Robinson tensor and quadratic forms in Riemann curvature 55
A.1 The definition of the Bel-Robinson tensor 55
A.2 The relationship between the Bel-Robinson tensor and energy in a small region 58
A.3 Quadratic form in Riemann curvature 59

參考文獻

References
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指導教授

聶斯特(J. Nester)

審核日期

2005-7-21

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