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研究生： 龔大勝
Da-Sheng Kung
論文名稱： SL(4,R)理論下的漸近平直對稱轉換
Asymptotic Flatness Preserving Transformations in SL(4,R) sigma-model
指導教授： 陳江梅
Chiang-Mei Chen
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： 96
語文別： 英文
論文頁數： 54
中文關鍵詞： 漸近平直 、SL(4 、R)模型 、五維黑洞
外文關鍵詞： asymptotic flat, SL(4, R) sigma-model, five dimensional black hole, black ring
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    • 考慮一個具有SL(4,R)對稱性的五維理論，其中包含重力場、一個純量場及一個三階向量場，我們提出保持時空漸近平直的對稱轉換所必須滿足的條件，統整出在不同座標描述的平直時空中，滿足該條件的所有轉換。這篇論文主要針對以下三種不同時空結構做討論：Kaluza-Klein黑洞、五維的黑洞和black ring，我們詳細列出所有滿足漸近平直條件的對稱轉換，並且討論這些轉換所代表的物理意義，其中部分轉換可以給出帶有電荷的解。

    We give a systematic method to determine the asymptotic flatness preserving transformations
    in the three-dimensional SL(4,R)/SO(2, 2) sigma-model arising from a
    five-dimensional gravity coupled to a dilaton and a three-form field. The permitted
    transformations depend on the coordinate choices. By focusing on three cases,
    namely the Kaluza-Klein black hole, five-dimensional black hole and black ring, we
    find out all possible asymptotic flatness preserving transformations and apply them
    to generate charge from single rotating vacuum solutions.

    1 Introduction 1 2 SL(4,R) Symmetry 4 2.1 Sigma-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Basic symmetry transformations . . . . . . . . . . . . . . . . . . . . . 7 2.3 Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3 Asymptotic Flatness Preserving 13 3.1 AFP condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Kaluza-Klein vacuum R3,1 × S1 . . . . . . . . . . . . . . . . . . . . . 15 3.3 5D Minkowski R4,1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.4 Ring coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4 Charged Kaluza-Klein Black Holes 21 4.1 R1 − L1 transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.2 T2 + T3 transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.3 R2 + L3 transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.4 S2 + S3 transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.5 R3 + L2 transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.6 R4 − L4 transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5 Charged 5D Black Holes 26 5.1 S3 transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.2 R2 + L3 transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.2.1 Physical quantity . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.2.2 An equivalent approach . . . . . . . . . . . . . . . . . . . . . 29 6 Charged Ring Solutions 31 6.1 Neutral black rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 6.2 S3 transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 6.3 R2 + L3 transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 34 6.3.1 Physical quantity . . . . . . . . . . . . . . . . . . . . . . . . . 35 6.3.2 An equivalent approach . . . . . . . . . . . . . . . . . . . . . 37 7 Conclusion 39 7.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 7.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Bibliography 41 A Kaluza-Klein theory 43 B Relations of gauge field components 45 C Black Rings 46 C.1 Ring coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 C.2 Neutral black rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 D Target-space potentials 52

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