臺灣博碩士論文加值系統

這本篇的論中，我們所要研究的主題是Bianchi type-IV and VIII 宇宙時空模型的物理現象。

要了解一個時空模型，最首要就是要先瞭解這時空基本的幾何量(i.e Riemann tensor，Ricci tensor，Ricci scalar)，在本篇論文的第一章，我們首先要計算出這些基本的幾何量，進一步我們使用愛因斯坦場方程式得到 Bianchi type-IV and VIII 的場方程式，求得場方程式的目的是為了解 Bianchi type-IV and VIII 宇宙時空模型隨時間演化的情形。

第二章，我們要探討的主題是變分法，要是我們直接對scalar factors變分得到的場方程式與使用愛因坦坦場方程公式所計算的場方程式有差異，解決的辦法是
Scalar factors中必須加入空間的變數，也就是說當我們在對 scalar factors 進行變分，必須要考慮空間部分的貢獻，這樣算出的結果才會與愛因斯坦場方程所得到的結果一致。

有許多的物理機制多有可能造成宇宙的膨脹，在第三章，我們討論純量場誘發宇宙膨脹的機制，我們所計算出的結果符合了inflation theory 中推論早期宇宙有一個很大的瞬間爆脹。

A primary subjects studied in this thesis are Bianchi type-IV and VIII cosmology models。

If one wants to understand the basic concepts of a spactime geometry，one should first know the Riemann tensor，Ricci tensor，Ricci scalar of the spactime geometry。
Chapter one in this thesis，we start to calculate the basic quantity of Bianchi type-IV and VIII cosmology models，furthermore，by using Einstein field equation we can derive the field equations of Bianchi type-IV and VIII cosmology models，with the solutions of these field equations we can understand the evolution of Bianchi type-IV and type-VIII cosmology models 。

In chapter two，what we discuss is variation method，if one derive the field equations by variating EH action with respect to a_i( t )，one will find out that the equations are different from the field equations which we derive from Einstein field equations。The method to solve this problem is to put the space variable into the
a¬_i(t)，and do the same calculation procedure as before，we find out the conclusion will consistent with the field equations which we derive from Einstein equation。

There are many physical mechanisms that will cause the universe expands，in chapter three，we canvass the scalar field is the physical mechanism that result in the universe to expand，the result of our calculation is consistent with the conclusion of inflation theory。

1 導言 3
2 Bianchi IV and VIII 模型 5
2.1 Bianchi IV and VIII 的基本計算. . . . . . . . . . . . . . . . . 5
2.2 高階曲率項. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Deriving field equation by Einstein equation . . . . . . . . . . 12
2.4 Bianchi identity . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 真空場的解. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.6 非真空態場方程的解. . . . . . . . . . . . . . . . . . . . . . . . 20
2.7 非均向性到均向性. . . . . . . . . . . . . . . . . . . . . . . . . 26
2.8 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 變分問題的探討 28
3.1 Field equation by variation method . . . . . . . . . . . . . . . 28
4 The induced gravity model 37
5 結論 42
6 附錄 43
6.1 使用的符號跟單位. . . . . . . . . . . . . . . . . . . . . . . . . 43
6.2 使用的公式. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.3 Spin connection . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.4 Curvature tensor . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.5 愛因斯坦場方程推導. . . . . . . . . . . . . . . . . . . . . . . . 48
6.6 Freidmann equation 的推導. . . . . . . . . . . . . . . . . . . . 53
6.7 ai field equation 推導. . . . . . . . . . . . . . . . . . . . . . . 54
6.8 Bianchi spacetimes models . . . . . . . . . . . . . . . . . . . . 57
6.9 The relation of scale factors in Bianchi models . . . . . . . . 61


List of Figures
2.1 Bianchi type-VIII model with Λ = 1,∆ = H2 − H1 . . . . . . . 2.2 Bianchi type-VIII model with Λ = 1 ,a1: 紅色線、a2: 綠色線, 19
本圖是使用Eq.2.37,Eq.2.39去做數值模擬. . . . . . . . . . . . 19
2.3 Bianchi type-VIII model with Λ = 1, 本圖是使用Eq.2.72,Eq.2.73
去做數值模擬. . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Bianchi type-VIII model with Λ = 1 . . . . . . . . . . . . . . . 25
4.1 Bianchi type-VIII model with Λ = 1,φ對時間的作圖, 本圖是使
用Eq.4.3,Eq.4.6去做數值模擬. . . . . . . . . . . . . . . . . . 39
4.2 Bianchi type-VIII model with Λ = 1,lna1: 紅色線,lna2: 綠色
線對時間的作圖, 本圖是使用Eq.4.3,Eq.4.6去做數值模擬. . . . 39

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