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研究生： 邱慕真
Mu-Chen Chiu
論文名稱：
Gravitational Lensing in Bekenstein''s Relativistic MOND
指導教授： 高仲明
Chung-Ming Ko
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 物理學系
Department of Physics
畢業學年度： 94
語文別： 英文
論文頁數： 82
外文關鍵詞： gravity, MOND, dark matter, gravitational lensing
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    • 有兩種不同的觀點能被用來解釋螺旋星系的光度與質量破缺。一者是引進暗物質的概念，一者則是從修改牛頓力學著手。 對於後一種的切入觀點，缺乏完整的相對性重力理論，一直是它無法令人信服的主要原因之一。 而這個破口終於在2004年，貝庚斯坦的“張量-向量-純量”理論出現之後，給填補起來了。 不同於之前力圖填補起這項遺缺的重力理論，貝庚斯坦的“張量-向量-純量”重力理論，能在不需要暗物質的前提下，完善地解釋重力透鏡系統，高彎曲角的現象。
    這篇論文，就是接續貝庚斯坦的工作，以他的“張量-向量-純量”重力理論文為基礎，去探究重力透鏡系統的許多現象，諸如：光彎曲、亮度放大與時間延遲。我們發現，在這個理論底下，光的彎曲角，會隨著最光與重力透鏡的最短距離的增加，最終趨於一個固定值。此外我們發現微重力透鏡的光變曲線，在傳統的廣力相對論和貝庚斯坦的“張量-向量-純量”重力理論底下是存在有差異的。還有，不同於一般的物質，純量場對光的整體路徑（也就是行進時間），起的是消減作用，而非增強。不幸的是，這個效應無法在重力透鏡的系統上，給予觀測證實。因為重力透鏡系統，所能觀測到的是同一光源的兩個影像到觀測者的時間差，而非單個影像的光到觀測者花費的時間。所幸的是，這個時間差，雖然無法用來證實純量場對時空的縮減效應，卻能連同光彎曲角的結果，來排除暗物質或修改牛頓力學的其中一個可能性。如果要求一個在廣義相對論底下的質量mgN和一個在貝庚斯坦的“張量-向量-純量”重力理論底下的質量mgM要產生先同的彎曲角，他們就不可能造成相同的時間延遲。

    Since Bekenstein''s (2004) creation of his Tensor-Vector-Scalar theory (TeVeS),
    the Modified Newtonian dynamics (MOND) paradigm
    has been redeemed from the embarrassment of lacking a relativistic version.
    One primary success of TeVeS is that it provides a
    satisfactory explanation of gravitational lensing without invoking dark matter,
    which could not be achieved by other MONDian theories.
    Following Bekenstein''s work, we investigate the phenomena of gravitational
    lensing including deflection angles, lens equations and time delay.
    We find that the deflection angle would maintain its value while the distance
    of closest approach vary in the MOND regime. We also use
    the deflection angle law to derive magnification and investigate microlensing light curves.
    We find that the difference in the magnification of the two images in the point mass model
    is not a constant such as that in GR. Besides, microlensing light
    curves could deviate significantly from GR in the deep MOND regime.
    Furthermore, the scalar field,
    which enhances the deflection angle in T$e$V$e,$S, contributes
    a negative effect on the potential time delay. Unfortunately this phenomenon is unmeasurable
    in lensing systems where we can only observe the time delay between two images
    for a given source. However, this kind of time delay (it is called measurable time
    delay in this thesis) offers another
    constraint
    on the mass ratio of the dark matter and MOND scenarios, which in general differs from that
    given by the deflection angle. In other words, for a lensing system, if two masses,
    m_{gN} and m_{gM}, are mutually alternatives for the deflection angles in their own paradigm,
    regarding the time delay they are in general exclusive.

    Contents 1 Introduction 1 2 Observational MOND 6 2.1 Spiral Galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.1 Tully-Fisher relation . . . . . . . . . . . . . . . . . . . 7 2.1.2 Rotation Curves Analysis . . . . . . . . . . . . . . . . 9 2.1.3 Two-body Relaxation and Dwarf Galaxies . . . . . . . 16 2.2 Elliptical Galaxies and Clusters of Galaxies . . . . . . . . . . . 17 2.2.1 Large Ellipticals . . . . . . . . . . . . . . . . . . . . . . 17 2.2.2 Clusters of Galaxies . . . . . . . . . . . . . . . . . . . . 17 2.3 Gravitational Lensing . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.1 Intuitional Approach to Strong Lensing . . . . . . . . 20 2.3.2 Weak Lensing . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.3 Microlensing, Statistic Lensing and Others . . . . . . . 25 2.4 Large Scale Structure and CMB . . . . . . . . . . . . . . . . . 27 2.4.1 CMB and Cosmology in MOND Paradigm . . . . . . . 27 2.4.2 Large Scale Structure . . . . . . . . . . . . . . . . . . . 30 3 Bekenstein''s Gravitation Theory 36 3.1 Steps to TeVe S . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.1.1 AQUAL . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.1.2 RAQUAL . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.1.3 Before the Sunrise - Sanders'' Strati¯ed Theory . . . . . 39 3.2 TeVe S: relativistic gravitational theory of MOND . . . . . . . 40 3.2.1 Actions of TeVe S . . . . . . . . . . . . . . . . . . . . . 41 3.3 The limits of TeVe S . . . . . . . . . . . . . . . . . . . . . . . 42 3.3.1 General relativity limit . . . . . . . . . . . . . . . . . . 43 3.3.2 Nonrelativistic limit . . . . . . . . . . . . . . . . . . . 45 3.4 PPN in TeVe S . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.5 Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4 Gravitational Lensing Phenomena in TeVe S 50 4.1 The Cosmological Model . . . . . . . . . . . . . . . . . . . . . 50 4.1.1 Framework of Gravitational Lensing . . . . . . . . . . . 50 4.1.2 The Cosmological Model . . . . . . . . . . . . . . . . . 51 4.2 Gravitational Lensing in TeVe S . . . . . . . . . . . . . . . . . 53 4.2.1 De°ection Angle in Terms of Mass . . . . . . . . . . . 53 4.2.2 Two Limits of the De°ection Angle . . . . . . . . . . . 56 4.2.3 Magni¯cation and Microlensing . . . . . . . . . . . . . 57 4.3 Time Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.3.1 Arrival Time in TeVe S . . . . . . . . . . . . . . . . . . 62 4.3.2 Arrival Time in the MOND Regime . . . . . . . . . . . 63 4.3.3 Measurable Time Delay . . . . . . . . . . . . . . . . . 64 4.3.4 Time Delay and Lensing Equation . . . . . . . . . . . . 69 5 Summery 71 5.1 Discussion & Summery . . . . . . . . . . . . . . . . . . . . . . 71

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