An optimal choice of reference for the quasi-local energy and angular momentum

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孫綱(Gang Sun) 查詢紙本館藏

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物理學系

論文名稱

(An optimal choice of reference for the quasi-local energy and angular momentum)

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摘要(中)

只要提供一個合適的演化向量和適當的背景幾何，哈氏量的邊界項能給出引力場準局
域的值。本文的目標在建構一套最佳化的辦法來確定適當的背景值。首先將封閉二維面
上度規的十個分量，從動力學時空等度規地嵌入到在背景幾何上，然後透過要求準局域
能量取極值的方法來確定適當的背景值。這套辦法也同時決定了演化向量的選取。我們
以軸對稱的動力學時空，針對克爾度規的情況明確地計算了準局域的能量和角動量。

摘要(英)

The boundary term of the gravitational Hamiltonian can be used to give the values
of the quasi-local quantities as long as one can provide a suitable evolution vector field and an
appropriate reference geometry. On the two-surface boundary of a region we have proposed using
four dimensional isometric matching between the dynamic spacetime and the reference geometry
along with energy extremization to find both the optimal reference matching and the appropriate
quasi-Killing vectors. Here we consider the axisymmetric spacetime case. For the Kerr metric
in particular we can explicitly solve the equations to find the best matched reference and quasi-
Killing vectors. This leads to the exact expression for the quasi-local boundary term and the values
of our optimal quasi-local energy and angular momentum.

關鍵字(中)

★ 哈氏量的邊界項
★ 準局域能量
★ 準局域角動量
★ 四維等度規配合

關鍵字(英)

論文目次

1 Prelude 1
1.1 Pseudotensor and superpotential . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Fundamental concepts of quasilocal . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Some applications of quasilocal . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.1 Tidal heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.2 Positivity of the gravitational energy . . . . . . . . . . . . . . . . . . . . 5
1.3.3 The cosmic censorship conjecture . . . . . . . . . . . . . . . . . . . . . 5
2 Conserved quantities in physics 6
2.1 Noether’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Conservation quantities in Minkowski space . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Matter field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 10 conserved quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Komar mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 The Covariant Hamiltonian Formalism 12
3.1 From Lagrangian to Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Refine the Hamiltonian boundary term . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 Quasilocal flux expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.4 Examples of quasilocal values in different spacetimes . . . . . . . . . . . . . . . 18
3.4.1 asymptotically flat to spatial infinity . . . . . . . . . . . . . . . . . . . . 18
3.4.2 asymptotically flat to null infinity . . . . . . . . . . . . . . . . . . . . . 19
3.4.3 small region limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.5 Remark and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 The optimal Hamiltonian boundary term 23
4.1 The adapted settings to fix reference configuration for several quasilocal expressions 23
4.2 The strategy to optimize reference and evolution vector . . . . . . . . . . . . . . 25
4.2.1 4D isometric embedding . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2.2 How to choose a suitable evolution vector . . . . . . . . . . . . . . . . . 28
4.2.3 Energy extremization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2.4 Search the critical point . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2.5 Examine the critical value . . . . . . . . . . . . . . . . . . . . . . . . . 34
5 Coda 37
5.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.1.1 The Optimal Quasilocal Energy . . . . . . . . . . . . . . . . . . . . . . 37
5.1.2 The Optimal Angular momentum . . . . . . . . . . . . . . . . . . . . . 39
5.2 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Appendix 45
Bibliography 48

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指導教授

聶斯特

審核日期

2013-7-22

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