準局部能量與參考系之選擇

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劉建良(Jian-Liang Liu) 查詢紙本館藏

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物理學系

論文名稱

準局部能量與參考系之選擇
(On quasi-local energy and the choice of reference)

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摘要(中)

根據協變哈密頓方法，我們可以決定引力系統的準局部量。有幾個相依於邊界條件的邊界項是可行的，但其中有一個 (與協變Dirichlet邊界條件相關)有最佳的性質；它給出ADM能量、Bondi能量及能流的正定性。此表示式一如其他表示式一樣也依賴於參考系及位移向量的選擇。如何做出最佳的選擇至今仍不清楚。本文計算幾個例子，包括FRW宇宙、第五類Bianchi模型、及Schwarzschild幾何在三種不同座標系的情形。計算結果顯示：準局部能量並不是唯一決定的，甚至於閔氏時空也能得到非零解。此結果是因為選擇不同參考系的緣故，而對於平直時空，度規張量仍有任意的選擇方式。我們藉由對準局部能量取極值的方式，找到一種決定參考系及位移向量的條件，使能量的表示式只依賴於物理系統，然而參考系的選擇仍有一定的任意性。此能量表示式在Schwarzschild幾何的三個不同形式下皆得到相同的值。

摘要(英)

Using the covariant Hamiltonian approach, we can determine the quasi-local quantities for a gravitating system within a region from an integral over its two-boundary. There are several possible boundary terms associated with different boundary conditions, but there is one (which corresponds to a kind of covariant Dirichlet condition on the metric) which has the best properties; it gives the ADM and Bondi energy and energy flux as well as having a positivity property. Like others this expression depends on the choice of reference configuration and the displacement vector field. It is not yet clear how to best make these choices. Here we calculate several cases including the FRW cosmologies, the Bianchi V model, and the Schwarzschild geometry in three different coordinate systems. The results imply that the quasi-local energy is not uniquely determined, even in Minkowski space one could also get a nonvanishing value. This result comes from the different choices of the reference frame. There is an arbitrary choice for the reference of flat space. By extremizing the quasi-local energy, we found a strategy to choose both the reference and the displacement vector to get the same energy value in the three Schwarzschild geometry cases. Although there are still many choices, it is not quite arbitrary anymore.

關鍵字(中)

★ 哈密頓
★ 邊界項
★ 準局部能量
★ 參考系
★ 位移向量

關鍵字(英)

★ displacement vector
★ Hamiltonian
★ reference
★ boundary term
★ quasi-local energy

論文目次

1 Introduction 1
2 Background 4
2.1 First-order Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Hamiltonian formulation . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Application to general relativity . . . . . . . . . . . . . . . . . . . . . 10
3 Calcuations 13
3.1 FRW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Bianchi V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Schwarzschild . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4 Eddington-Finkelstein . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5 Painlev′e-Gullstrand . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.6 Extremize the energy to restrict the reference . . . . . . . . . . . . . 29
3.6.1 FRW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.6.2 Schwarzschild . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.6.3 Eddington-Finkelstein . . . . . . . . . . . . . . . . . . . . . . 34
3.6.4 Painlev′e-Gullstrand . . . . . . . . . . . . . . . . . . . . . . . . 34
4 Conclusion 36
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
A Variation of Einstein-Hilbert action 42
B Mean curvature 45
B.1 FRW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
B.2 Bianchi V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
B.3 Schwarzschild . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
B.4 Eddington-Finkelstein . . . . . . . . . . . . . . . . . . . . . . . . . . 49
B.5 Painlev′e-Gullstrand . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
C Different reference frames 51
C.1 Schwarzschild . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
C.2 Eddington-Finkelstein . . . . . . . . . . . . . . . . . . . . . . . . . . 55
C.3 Painlev′e-Gullstrand . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
C.4 Another choice of reference frames . . . . . . . . . . . . . . . . . . . . 60
C.4.1 Schwarzschild . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
C.4.2 Eddington-Finkelstein . . . . . . . . . . . . . . . . . . . . . . 63
C.4.3 Painlev′e-Gullstrand . . . . . . . . . . . . . . . . . . . . . . . . 65

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指導教授

聶斯特(James M. Nester)

審核日期

2007-7-19

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