這篇論文主要的想法是把我們所存在的三維空間，假設成被埋藏在五維球對稱時空中的三維球膜，我們利用外部區率[extrinsic curvature]和 Israel junction condition得到球膜的運動方程式。我們探討了物質主宰和輻射主宰兩種不同的流體之後，發現宇宙演化的方式和宇宙常數有非常大的關係，根據不同的宇宙常數我們把宇宙的演化分成四類。接著我們把宇宙常數假設成高維時空中的位能來探討球膜演化的方式，最後我們簡述了兩層球膜的狀況。

The main idea of this thesis is to image the three-dimensional space as a three-brane embedded in a five-dimensional spherical symmetry spacetime.  We get the equation of motion from the extrinsic curvature and Israel junction condition.  After discussing the equation of motion in dust case and radiation dominated case, we know that the cosmological constants dominate the behavior of our model.  According to different pair of cosmological constants, we divide the evolution of our model into four types.  We suppose the cosmological constant as the potential energy in five-dimensional spacetime to analyze the evolution of our model.  Finally, we give a suggestion about two branes system.

Chapter 1
Introduction

1.1Extra dimensions…………………………………………………1
1.2Spherical brane …………………………………………………4

Chapter 2
The equation of motion of a (3+1)-dimensional spherical brane

2.1 5-dimensional Schwarzschild-like solution………………6
2.2 Extrinsic curvature……………………………………………8
2.3 Equation of motion  …………………………………………12

Chapter 3
The evolution of the (3+1)-dimensional spherical brane

3.1 w=0, dust case…………………………………………………16
3.2 w=1/3, radiation dominated case …………………………28
3.3 Summary …………………………………………………………31


Chapter 4
Discussion

4.1 Cosmological constants………………………………………34
4.2 Two 3-branes embed in five-dimensional space time …37

Reference


Figures and Tables
Figure 1.1: A three brane embedded in a five dimensional spacetime….3
Figure 1.2: A three shell brane embedded in a five dimensional spherical
symmetry spacetime…………………...…………………4
Figure 3.1: The evolution when the cosmological constants are zero on
the both sides of the brane in dust case……………………17
Table 3.1: Some example with specific values of Λ………………...18
Figure 3.2: The evolution before the maximum radius as ± Λ being 64 27
on both sides in dust case……..................................19
Figure 3.3: The acceleration of Λ+=Λ− in dust case……….…..………..20
Figure 3.4: Effective potential of Λ+=Λ− in dust case………………….20
Figure 3.5: The figures of Λ+=Λ- and Λ<0 in dust case……………..21
Figure 3.6: The figures of | | | | + − Λ >> Λ and (Λ Λ ) = (+ +) (+ −) + − , , , , in dust
case..…………………………………………………..23
Figure 3.7: The eff V when Λ = 10 + , | | | | + − Λ >> Λ and (Λ Λ ) = (+ +) (+ −) + − , , , ,
in dust case.………………………………………………..23
Figure 3.8: The figures of | | | | + − Λ >> Λ and (Λ Λ ) = (− +) (− −) + − , , , , in dust
case.………………………………………………………..24
Figure 3.9: The eff V whenΛ = 10 + , | | | | + − Λ >> Λ and (Λ Λ ) = (− +) (− −) + − , , , ,
in dust case………………………………………………...25
Figure 3.10: The figures of | | | | + − Λ << Λ and (Λ Λ ) = (+ +) (− +) + − , , , , in
dust case.……………………………….…………………..26
Figure 3.11: The figures of | | | | + − Λ << Λ and (Λ Λ ) = (+ −) (− −) + − , , , , in dust
case.………..……………………………………………..27
Figure 3.12: Three different Λ± of radiation dominated case………..…29
Figure 3.13: The velocity of radiation dominated case………...………30
Figure 3.14: Four types of the evolution in our models…………..……31
Figure 3.15: The different universe types with different pair of the
cosmological constants in the dust case.…………………32
Figure 3.16: The different universe types with different pair of the
cosmological constants in the radiation dominated case…33
Figure 4.1: The evolutions of our model with the assuming on
cosmological constants.……………………………………36
Figure 4.2: Two three shell branes embedded in a five dimensional
spherical symmetry spacetime……………………………..37

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