臺灣博碩士論文加值系統

本論文由John D. Barrow一篇非均向暴漲宇宙出發，先探討Bianchi type II & VIh 在考慮了曲率二次修正項的Lagrangian density中，透過直接解其場方程式，計算出其能量動量張量為何。然而，由於違背了能量條件，因此我們嘗試引入膜宇宙學的概念，探討四維時空是一個鑲嵌在五維時空中的膜時，這多出來的一個多餘維度會使場方程式改變，最後再計算在這兩種時空結構下其能量動量張量為何。

In this thesis, we start at John D. Barrow’s paper, the “Anisotropically inflating universes”. We discuss the Bianchi type II & VIh for the Lagrangian density with the quadratic curvature terms, and calculate the energy-momentum tensor by solving the field equation. However, the energy condition is violated. So we introduce the concept of brane and discuss the extra dimension affect the field equation when we consider the 3-brane world. Finally we calculate the energy-momentum tensor of these two cosmology models once again.

1 緒論
1.1 宇宙論 . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 研究動機 . . . . . . . . . . . . . . . . . . . . . . 1
2 Bianchi type model 3
2.1 Bianchi Group . . . . . . . . . . . . . . . . . . . 3
2.2 Bianchi classcation . . . . . . . . . . . . . . . . 3
2.3 Killing equation . . . . . . . . . . . . . . . . . . 4
2.4 三維下的九種G3運動群 . . . . . . . . . . . . . . . . 4
2.5 Bianchi type II 的度規 . . . . . . . . . . . . . . . 5
2.6 Bianchi type VIh的度規 . . . . . . . . . . . . . . . 7
3 膜上的等效愛因斯坦方程式 9
3.1 Braneworld . . . . . . . . . . . . . . . . . . . . . 9
3.2 Effective 4-dim gravitation equations of the brane . 9
4 Bianchi type II,VIh models 13
4.1 Bianchi type II . . . . . . . . . . . . . . . . . . 13
4.1.1 metric . . . . . . . . . . . . . . . . .. . . . . 13
4.1.2 計算場方程式的解 . . . . . . . . . . . . . . . . 14
4.1.3 引入膜宇宙學之概念. . . . . . . . . . . . . . . . 18
4.1.4 解出能量動量張量. . . . . . . . . . . . . . . . . 20
4.2 化簡brane equation. . . . . . . . . . . . . . . . . 21
4.2.1 驗證其解相同. . . . . . . . . . . . . . . . . . . 22
4.2.2 能量動量張量守恆. . . . . . . . . . . . . . . . . 24
4.2.3 解1 . . . . . . . . . . . . . . . . . . . . . . . 25
4.2.4 解2 . . . . . . . . . . . . . . . . . . . . . . . 25
4.3 Bianchi type V Ih 模型. . . . . . . . . . . . . . . 25
4.3.1 度規. . . . . . . . . . . . . . . . . . . . . . . 25
4.3.2 brane equation . . . . . . . . . . . . . . . . . 26
4.3.3 計算方程式中各項. . . . . . . . . . . . . . . . . 26
4.3.4 解出能量動量張量. . . . . . . . . . . . . . . . . 27
4.3.5 改用化簡方法處理Bianchi type VIh 模型 . . . . . . 28
4.4 結論. . . . . . . . . . . . . . . . . . . . . . . . 30
A 符號定義 31
B curvature的計算結果 32
B.1 Bianchi type II model . . . . . . . . . . . . . . . 32
B.2 Bianchi type VIh model. . . . . . . . . . . . . . . 34
C 各Bianchi type model 的原始度規 36
D 單位 37

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