臺灣博碩士論文加值系統

根據哈伯的觀測,宇宙是一個動態的宇宙, 因此,瞭解宇宙是如何從最初始的狀態演化成現在的狀態，就是一個很重要的問題。

如果宇宙無毛猜想是正確的，所有不均向膨脹的宇宙最後都會演化成 de Sitter 時空，但是這個猜想只得到部分證實。

在先前的論文 [1] 中, 在由里奇張量跟里奇純量構成的二次項的重力理論裡，已經找到一組Bianchi type I 時空的不均向膨脹解，我們將在論文中呈現正確的解。我們可以用對場方程式做微擾的方式，證實這個解是不穩定的。但是這篇論文也將以論文 [1] 的動態系統分析方式，重現這個不均向膨脹解的穩定性探討，並探討這兩者之間的關聯 。

The evolution detail of an anisotropically expanding universe is an interesting research focus lately. The no hair conjecture, proved partially by Robert Wald, states that all anisotropically expanding universes will tend to an isotropic de Sitter space.

A class of anisotropically expanding solutions are found in ref. [1] for a gravity model with second-order correction terms derived from all possible combinations of the Ricci curvature tensor and Ricci scalar. We will present the correct expanding solutions in this paper. These solutions can be shown to be unstable by perturbing the field equations. Conventional approach by dynamical system analysis used in ref. [1] will be reviewed carefully and compared with the perturbation method mentioned earlier in this paper.

Chapter 1 介紹 1
1.1 背景 1
2.2 廣義相對論概述 2
1.2.1 基本觀念 2
1.2.2 廣義相對論 3
1.3 宇宙無毛定理 4
1.4 動量和能量 5
1.5 Friedmann Robertson-Walker 度規 6
1.6 De Sitter 時空 7
Chapter 2 Bianchi 模型 8
2.1 序言 8
2.2 李代數和李群 8
2.3 Killing 方程式 9
2.4 G3 的 Motions 9
2.5 Bianchi 型態 I 10
2.6 九種 Bianchi 型態 12
Chapter 3 Bianchi 型態 I 模型 13
3.1 前言 13
3.2 基本計算 14
3.2.1 克里斯托夫符號 (Christoffel Components) 14
3.2.2 黎曼張量 (Riemann tensor) 15
3.2.3 里奇張量 (Ricci tensor) 15
3.2.4 里奇純量 (Ricci tensor) 16
3.2.5 愛因斯坦張量 (Einstein tensor) 16
3.3 場方程式的推導 16
3.3.1 對尺度因子 (Scale factors) 做變分 16
3.3.2 對g^uv 做變分 20
3.3.3 場方程式 23
3.4 膨脹解 (Inflating solutions) 23
3.5 能量條件 25
Chapter 4 穩定性 27
4.1 微擾 28
4.2 穩定性分析 30
Chapter 5 動態系統方法 32
5.1 動態系統 (Dynamical systems) 32
5.2 簡介 33
5.3 基本運算 34
5.4 場方程式 35
5.5 膨脹解 39
5.6 比較:回到原始的變數，哈柏參數 40
5.7 穩定性 48
Chapter 6 微擾方法跟動態系統的連結 71
6.1 特徵值的轉換 71
6.2 基本原理 71
Chapter 7 結論
A 對 g^uv 做變分 77
B 微擾 82
C 誘發重力 (Induced gravity) 88
C.1 爆漲 (Inflation) 88
C.2 純量場的誘發重力 88
C.2.1 對尺度因子 (Scale factors) 做變分 88
C.2.2 對 g^uv 做變分 91

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