在Kerr幾何的特殊正交座標系和狄拉克旋子

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楊寶鴻(Poh-Hoong Yong) 查詢紙本館藏

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物理學系

論文名稱

在Kerr幾何的特殊正交座標系和狄拉克旋子
(Special orthonormal frames and Dirac spinors in Kerr geometry)

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摘要(中)

愛因斯坦的廣義相對論是一個與座標選取無關的理論，所以我們有選取座標系的自由度。我運用聶斯特教授發展出來的規範條件來選取一個正交歸一的座標，並運用在Reissner-Nordstrom 和 Kerr 幾何時空上面。這個規範條件選擇了一個特定的標架。這種特定的標架在重力系統的正能量定理証明上特別有用。另外，這種特定的標架和狄拉克方程試息息相關，透過解狄拉克方程，我們就可以選出特定的標架滿足規範條件。然而，在彎曲的時空中，要解出狄拉克方程不是一件簡單的事。我們考慮了弱重力場條件下如何定義特定的標架，還有在Reissner-Nordstrom 和 Kerr 時空上求出狄拉克方程漸近常數的解。

摘要(英)

Since Einstein’’s gravity theory is a frame independent theory, we have the freedom of choosing an orthonormal frame. I use Nester’’s gauge condition to select a preferred orthonormal frame for some gravitational systems including the Reissner-Nordstrom and Kerr geometry. The gauge condition selects a special orthonormal frame (SOF). A SOF has application in particular to a positive energy proof and energy localization for a gravitational system. This gauge condition is related to the solution of the Dirac equation; by solving the Dirac equation we can determine a special orthonormal frame. However, in curved spacetime the solution of Dirac equation is highly nontrivial. We calculated the weak field limit case and
found the asymptotically constant solution for the Dirac equation in the Reissner-Nordstrom and Kerr geometry.

關鍵字(中)

★ 狄拉克
★ 旋子
★ 座標系

關鍵字(英)

★ Dirac spinor
★ orthonormal frame
★ Kerr geometry

論文目次

1 Introduction 1
2 Special orthonormal frames 4
2.1 Orthonormal tetrads. . . . . . . . . . . . . . . . 4
2.2 Special orthonormal frames . . . . . . . . . . . . 5
2.3 Isotropic metric . . . . . . . . . . . . . . . . . 10
3 SOF in Reissner-Nordstrom geometry 12
3.1 Isotropic form of Reissner-Nordstrom . . . . . . . 12
3.2 Three-dimensional R-N solution. . . . . . . . . . . 14
4 Dirac spinor in Kerr geometry 23
4.1 Weak field approximation of the Kerr solution. . . . 24
4.2 Newman-Penrosre formalisms . . . . . . . . . . . . 26
4.3 Dirac equation in Kerr geometry . . . . .. . . . . . 27
4.3.1 Minkowski . . . . . . . . . . . . . . . . . . . . 31
4.3.2 Schwarzschild . . . . . . . . . . . . . . . . . . 32
4.3.4 Kerr . . . . . . . . . . . . . . . . . . . . . . .34
4.3.4 Frobenius method. . . . . . . . . . . . . . . . . 36
4.4 Scalar and psuedoscalar . . . . .. . . . . . . . .. 42
5 Conclusion 44
Bibliography . . . . . . . . . . . . . . . . . . . . 46

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指導教授

聶斯特(James M. Nester)

審核日期

2008-7-6

推文