Introduction to 2-spinors in general relativity

Preface

1 Spinor geometry 1
1.1 Minkowski space 1
1.2 The null cone and Riemann sphere 4
1.3 Spin transformations and spin matrices 7
1.4 Flagpoles and flag pianes 8
1.5 Spin-space 10
1.6 Exercises 11

2 Spinor algebra 13
2.1 Abstract index notation 13
2.2 Complex conjugation of spinor components 15
2.3 Vector bases and abstract indices 17
2.4 Levi-Civita spinor 18
2.5 Spinor dyad basis and its components 20
2.6 Spinor symmetry operations 22
2.7 The connection between world-tensors and spinors 24
2.7.1 Infeld-van der Waerden symbols 24
2.7.2 Null vectors 26
2.8 The decomposition of spinors 29
2.8.1 Spinor equivalent of a symmetric valence two tensor 30
2.8.2 Spinor equivalent of the electromagnetic field tensor 31
2.9 The canonical decoraposition of symmetric spinors 33
2.9.1 Classification of the electromagnetic spinor 34
2.9.2 Classification of the Weyl spinor 34
2.10 Exercises 37

3 Spinor analysis 39
3.1 Spinor form of the covariant derivative 39
3.2 The curvature spinors 41
3.2.1 Symmetries of the curvature spinors 42
3.2.2 The Ricci spinor 44
3.2.3 The Weyl spinor 46
3.3 Spinor equivalent of the Ricci identities 48
3.4 Spinor equivalent of the Bianchi identities 50
3.5 The Newman-Penrose spin coefficient formalism 52
3.5.1 Spin coefficients 52
3.5.2 The Newman-Penrose field equations 58
3.6 Newman-Penrose quantities under Lorentz transformations 64
3.6.1 Null rotation with 1 fixed 65
3.6.2 Null rotation with n fixed 66
3.6.3 Lorentz spin-boost transformation 67
3.7 Miscellaneous transformations 69
3.7.1 1 <-> n and m <-> —m 69
3.7.2 Prime operation 71
3.7.3 Asterisk operation 73
3.8 Geroch-Held-Penrose formalism 75
3.8.1 Spin- and boost-weighted scalars 76
3.8.2 Weighted and unweighted spin coefficients 77
3.8.3 Weighted differential operators 80
3.8.4 The Geroch-Held-Penrose field equations 82
3.8.5 A brief note on the modified GHP formalism 86
3.9 Goldberg-Sachs theorem 87
3.10 Exercises 89

4 Lanczos spinor 91
4.1 Introduction 91
4.2 Lanczos' Lagrangian 92
4.3 Lanczos' gauge conditions 97
4.4 The Lanczos spinor 100
4.5 The spinor version of the Weyl-Lanczos equations 102
4.6 The Lanczos coefficients 104
4.7 The Weyl-Lanczos equations in spin coefficient form 105
4.8 The Ricci-Lanczos equations in spin coefficient form 106
4.9 The behaviour of Lanczos coefficients under Lorentz transformations 109
4.9.1 Null rotation with 1 fixed 109
4.9.2 Null rotation with n fixed 110
4.9.3 Spin-boost transformation 111
4.10 Miscellaneous transformations 112
4.10.1 l <-> n and m <-> —m 112
4.10.2 Prime operation 112
4.10.3 Asterisk operation 113
4.11 The Weyl-Lanczos equations in GHP form 114
4.12 Solutions of the Weyl-Lanczos equations 114
4.12.1 Petrov type O space-times 115
4.12.2 Petrov type N space-times 116
4.12.3 Petrov type III space-times 116
4.12.4 Lanczos coefficients for the Schwarzschild metric 117
4.13 A brief note on the Lanczos spinor/tensor 120
4.14 Exercises 120

Appendix A: Aspects of general relativity 127
A.1 The space-time of general relativity 127
A.1.1 Space-time as a differentiate manifold 127
A.1.2 The tangent space Tpo (M) at a point Po of a manifold M 128
A.1.3 Tensor algebra 134
A.1.4 Tensor detection 138
A.1.5 An illustrative example from classical mechanics 140
A.1.6 Tensor fields 142
A.1.7 An illustrative example from special relativity 144
A.2 Riemannian geometry and tensor analysis 146
A.2.1 Introduction 146
A.2.2 Covariant derivatives 151
A.2.3 The curvature tensor 154
A.2.4 The Bianchi identities 157
A.2.5 Riemannian geometry and geodesies 158
A.3 General relativity 161
A.3.1 The field equations 161
A.3.2 The Newtonian approximation 164
A.3.3 The Schwarzschild solution 168
A.3.4 Planetary orbits 170
A.3.5 The deflexion of light 173
A.3.6 The gravitational red shift 175
A.4 Exercises 176
Bibliography 181
Index 185