多種證據表明:目前宇宙空間在大尺度上是均勻各向同性的。空間的形狀以度規張量來表示，也就是要求度規張量不隨空間點而改變；也不隨方向而改變。雖然當前的宇宙大尺度是均勻各向同性的，卻不能排除早期宇宙可能的不均勻不各向同性。本篇最終介紹了均勻不各向同性的宇宙。這種模型的前提假設是：宇宙在其空間每一點的度規都一樣，在空間的每個方向度規不一定一樣。其中又分為滿足四維對稱群和三維對稱群的形式，分別稱作Kantowski－Sachs models和Bianchi metric。要描述這類宇宙需要用到的基礎理論是廣義相對論和群論，我在目前看到的資料中普遍發現對這兩種理論的介紹比較抽象。例如協變微分、李導數：它們都有其直觀的意義，雖然這兩個概念的創立不需要藉助高維卡氏坐標，但創立這些概念的人一定是先藉助三維空間下觀察二維彎曲空間的形態而產生靈感的。另外，擁有直觀的感受更容易學習和教授，本篇在介紹基礎理論也是盡量直觀，避免複雜的數學。

A variety of evidence shows that:Space is homogeneous isotropic on a large scale. Since the shape of the space is expressed in terms of a metric tensor, it means that the metric tensor does not change with the position and direction. Although the current large-scale universe is homogeneous and isotropic, it cannot exclude the possible non-isotropy of the early universe. This thesis introduced a homogeneous but anisotropic universe. The assumption of this model is that the Universe has the same metrics at every point in its space and that the metrics in each direction of space are not necessarily the same. It can satisfy four-dimensional symmetry groups and three-dimensional symmetry groups, which are called Kantowski-Sachs models and Bianchi metrics,respectively. The basic theory that needs to be used to describe this type of universe is general relativity and group theory. I have found in recent textbooks that the introduction of these two theories is rather abstract. For example, covariant derivative, Lie derivative: they all have their intuitive meaning. Although the creation of these two concepts does not require the use of higher-dimensional Cartesian coordinates, the inspiration must have come up when people observe the two-dimensional curved space in three-dimensional flat space. In addition, having intuitive sensations makes it easier to learn and teach. The basic theory involved in this thesis is made as intuitive as possible.

介紹	1
第一章 廣義相對論簡介	2
1.協變導數	2
2.黎曼幾何	5
（1）測地線（geodesic）	5
（2）黎曼曲率（Riemann curvature）	6
（3）里奇張量（Ricci tensor）	9
3．愛因斯坦場方程	11
第二章 群	13
1.李導數	13
2.李群李代數	17
第三章 均勻宇宙	23
1．均勻各向同性宇宙	23
2．均勻不各向同性的宇宙	25
（1）Kantowski－Sachs models	26
（2）Bianchi metric	28
結論	32
參考文獻	33

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