在本論文內，我們考慮了高維度史瓦西黑洞時空中的重力微子微擾場。在研究重力微子微擾的課題中，通常會引進 Newman-Penrose 方法，不過這個方法只適用在四維的時空中，而不能直接推廣到高維度時空，所以我們採用了另外一種較為直接的觀點來研究這個課題。我們首先研究旋量-向量場在N維球膜中的本徵值問題，並得到一個完整態。利用這些本徵值與本徵態分離了維度史瓦西黑洞時空中的重力微子微擾運動方程式的球對稱部分，並定義了運動方程式中的不變量，進而得到徑向方程式。我們討論了徑向方程式中的有效位勢，並研究了重力微子在高維史瓦西黑洞時空中的吸收機率以及準正則模的相關課題。

In this thesis we consider the gravitino perturbation on a general dimensional Schwarzschild black hole spacetime.  The analysis of gravitino fields in curved spacetimes is usually carried out by using the Newman-Penrose formalism, which is useful in four dimensional cases but cannot be generalized to higher dimensions in a straightforward manner.  We consider a more direct approach to derive the radial equations.  We start this study by finding a complete set of spinor-vector eigenmodes on an N-sphere which includes the "non TT eigenmodes" and the "TT eigenmodes".  We separate the angular part of the gravitino equations of motion by these eigenmodes.  With the consideration of the gauge symmetry, we write down the Schrödinger like radial equations for the gauge invariant variables.  We then discuss the effective potentials.  Lastly, we obtain the quasinormal modes and the absorption probabilities of the gravitino field in D dimensional Schwarzschild black hole spacetimes.

Contents
Abstract i
Acknowledgments iii
Contents iv
List of Tables vi
List of Figures vii
Chapter 1: Introduction 1
1.1 Quantum  eld theory in curved spacetime . . . . . . . . . . . . . . . 1
1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Organization to the thesis . . . . . . . . . . . . . . . . . . . . . . . . 5
Chapter 2: The eigenmodes of spin  elds on spheres 7
2.1 Notation on SN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Brief review of spinor  elds on SN . . . . . . . . . . . . . . . . . . . . 10
2.2.1 Eigenmodes on S2 . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.2 Odd N cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.3 Even N cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Overview of the spinor-vector eigenmodes on SN . . . . . . . . . . . . 16
2.4 Spinor-vector non TT eigenmodes on SN . . . . . . . . . . . . . . . . 18
2.5 Spinor-vector TT eigenmode I on SN . . . . . . . . . . . . . . . . . . 20
2.5.1 Odd N cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5.2 Even N cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.6 Spinor-vector TT Modes II on SN . . . . . . . . . . . . . . . . . . . . 26
2.6.1 Odd N cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.6.2 Even N cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Chapter 3: The radial equation of the gravitino  eld on a D-dimensional
Schwarzschild black hole spacetime 28
3.1 Massless Rarita-Schwinger equation . . . . . . . . . . . . . . . . . . . 29
3.2 The radial equation related to the non TT eigenmodes on SN . . . . 32
3.2.1 The equation of motion . . . . . . . . . . . . . . . . . . . . . . 32
3.2.2 Gauge invariant variable . . . . . . . . . . . . . . . . . . . . . 37
3.2.3 The radial equation . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.4 The e ective potential . . . . . . . . . . . . . . . . . . . . . . 43
3.3 The radial equation related to the TT eigenmodes on SN . . . . . . . 48
3.3.1 The equation of motion . . . . . . . . . . . . . . . . . . . . . . 48
3.3.2 The radial equation . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3.3 The e ective potential . . . . . . . . . . . . . . . . . . . . . . 52
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Chapter 4: Gravitino absorption probability in the D dimensional
Schwarzschild spacetime 54
4.1 Low energy region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1.1 For the spinor perturbation potential . . . . . . . . . . . . . . 55
4.1.2 For the spinor-vector perturbation potential . . . . . . . . . . 58
4.2 WKB approximation for the general energy region . . . . . . . . . . . 59
4.2.1 For the spinor perturbation potential . . . . . . . . . . . . . . 60
4.2.2 For the spinor-vector perturbation potential . . . . . . . . . . 64
Chapter 5: Quasi normal mode 66
5.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.2 The QNM for the gravitino  eld with spinor perturbation potentials . 68
5.3 The QNM for the gravitino  eld with spinor-vector perturbation potentials
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Chapter 6: Conclusions 76
Chapter A: The Gauge symmetry of the (A)dS Schwarzschild space-
time 80
Bibliography 83

List of Tables
5.1 Low-lying (n   l, with l = j3=2, and D = 4; 5) gravitino quasinormal
mode frequencies using the WKB methods on the spinor perturbation
potential V1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2 Low-lying (n   l, with l = j3=2, and D = 6; 7) gravitino quasinormal
mode frequencies using the WKB methods on the spinor perturbation
potential V1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.3 Low-lying (n   l, with l = j3=2, and D = 8; 9) gravitino quasinormal
mode frequencies using the WKB methods on the spinor perturbation
potential V1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.4 Low-lying (n   l, with l = j  1=2, and D = 5; 6; 7; 8) gravitino
quasinormal mode frequencies using the WKB methods on the spinorvector
perturbation potential V1. . . . . . . . . . . . . . . . . . . . . 75
List of Figures
3.1 The gravitino e ective potentials V1;2 for j = 1
2 , D = 5 to D = 8, the
spinor perturbation case. . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2 The gravitino e ective potentials V1;2 when D = 5, j = 1
2 in the spinor
perturbation case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3 The gravitino e ective potentials V1;2 for j = 3
2 , D = 4 to D = 8, the
spinor perturbation case. . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 The gravitino e ective potentials V1;2 for j = 3
2 , D = 9 to D = 11, the
spinor perturbation case. . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.5 The gravitino e ective potentials V2 for j = 3
2 , D = 21 to D = 22, the
spinor perturbation case. . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.6 The gravitino e ective potentials for higher D when j   5
2 , the spinor
perturbation case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.7 The gravitino e ective potentials V1;2 for j = 1
2 , D = 4 to D = 9, the
spinor-vector perturbation case. . . . . . . . . . . . . . . . . . . . . . 53
3.8 The gravitino e ective potentials V1;2 for j = 3
2 , D = 4 to D = 9, the
spinor-vector perturbation case. . . . . . . . . . . . . . . . . . . . . . 53
4.1 Gravitino e ective potentials and absorption probabilities in the 4D
Schwarzschild spacetime, spinor perturbation case. . . . . . . . . . . . 61
4.2 Gravitino e ective potentials and absorption probabilities in the higher
dimensional Schwarzschild spacetime for j = 1
2 , spinor perturbation case. 62
4.3 Gravitino e ective potentials and absorption probabilities in the 5D
Schwarzschild spacetime, spinor perturbation case. . . . . . . . . . . . 63
4.4 Gravitino e ective potentials and absorption probabilities in the 6D
Schwarzschild spacetime, spinor perturbation case. . . . . . . . . . . . 63
4.5 Gravitino absorption probabilities in the higher dimensional Schwarzschild
spacetime, spinor perturbation case. . . . . . . . . . . . . . . . . . . . 64
4.6 Gravitino e ective potentials and absorption probabilities on the 5D
Schwarzschild spacetime, spinor-vector perturbation case. . . . . . . . 65
4.7 Gravitino e ective potentials and absorption probabilities on the 6D
Schwarzschild spacetime, spinor-vector perturbation case. . . . . . . . 65
5.1 Gravitino QNM frequencies, spinor perturbation case. . . . . . . . . . 69
5.2 Variation of the QNM frequencies with dimensions for  x l and n = 0. 70
5.3 Gravitino QNM frequencies, spinor-vector perturbation case. . . . . . 74
viii

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