藉由研究光受到重力作用而產生軌跡偏轉的現象，討論由暗物質造成的重力系統，以不同的衝擊因子發射光子且使其經過暗物質星體所構成的區域時，偏轉角會產生變化的趨勢。尤其是光與暗物質間僅有重力作用，因此甚至可以討論當衝擊因子小於暗物質星的半徑，也就是光穿越暗物質星體的情況。首先探討的是靜態的情況，偏轉角如同預期有著對稱的分佈；此外當暗物質星體劇烈的旋轉時偏轉角的分佈圖型呈現符合座標拖曳必然的預期結果。然而當星體的旋轉方向與光子的傳遞方向相反時，偏轉的現象將會變得不穩定，這個部分需要再進一步的研究。

A self-gravitating system is setup to study the effect induced by dark matter. We consider a star of a solar mass formed by dark matter and explore the range of the deflection angle when a photon passing through the neighborhood of such a dark star for various impact factors. In particular, since the light interacts with the dark matter only gravitationally, impact factors with a size smaller than the radius of the dark matter star are allowed. When the dark star is static, the distribution of the deflection angle is symmetric as expected. On the other hand, the distribution pattern fits the essential anticipation of frame draggings when the dark star spins drastically. However, the phenomenon becomes unstable when the spinning direction of the dark star is opposite to the motion of the photon and needs further investigation.

Abstract………………………………………………………………………………………i
Content………………………………………………………………………………………iii
List of Figures..………………………………………………………………………………v
1 Introduction…………………………………………………………………1
1.1 Configuration of Deflection Angle……………………………………………2
2 Deflection of light near a Schwarzschild dark matter star ………………4
2.1 Deflection angle inside a Schwarzschild star…………………………………4
2.1.1 Exterior Solution…………………………………………………………4
2.1.2 Interior Solution……………………………………………………………5
2.2 Consistency Check on the Boundary…………………………………………6
3 Deflection of a Kerr dark matter star……………………………………8
3.1 Approach 1(Numerical) ………………………………………………………9
3.1.1 The Christoffel symbols…………………………………………………10
3.1.2 Geodesic Equation………………………………………………………11
3.1.3 Finding the relation of dr/dτ and dϕ/dτ …………………………………….14
3.2 Approach2(Analytical) ………………………………………………………17
3.2.1 Consistency Check with zero angular momentum…………………18
3.2.2 Results of Analytical Solution…………………………………………20
3.2.3 Deflection angle with parallel and anti-parallel angular momentum……20
3.2.4 Comparing the difference with parallel and anti-parallel angular
momentum in the exterior region……………………………………22
3.2.5 Comparing the difference with parallel and anti-parallel angular
momentum in the interior region……………………………………23
3.2.6 Conclusion of Analytical Solutions……………………………………23
4 Summary and Outlooks……………………………………………………24
4.1 Summary……………………………………………………………………………24
4.2 Outlooks……………………………………………………………………………24
Bibliography…………………………………………………………………………26

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