Preface
1 Rotations and Spinors
1.1 Representation of the rotations
1.2 Spinors
1.3 Elementary applications
1.4 Spinors in spaces with indefinite metric
2 Spin-Weighted Spherical Harmonics
2.1 Spherical harmonics
2.2 Spin weight
2.3 Wigner functions
3 Spin-Weighted Spherical Harmonics. Applications
3.1 Solution of the vector Helmholtz equation
3.2 The source-free electromagnetic field
3.3 The equation for elastic waves in an isotropic medium
3.4 The Weyl neutrino equation
3.5 The Dirac equation
3.6 The spin-2 Helmholtz equation
3.7 Linearized Einstein theory
3.8 Magnetic monopole
4 Spin-Weighted Cylindrical Harmonics
4.1 Definitions and basic properties
4.2 Representation of the Euclidean group of the plane
4.3 Applications
4.3.1 Solution of the vector Helmholtz equation
4.3.2 Elastic waves in an isotropic elastic medium
4.3.3 Solution of the equations of equilibrium for an isotropic elastic medium
4.3.4 Solution of the Dirac equation
4.3.5 Solution of the spin-2 Helmholtz equation
4.4 Parabolic and elliptic coordinates
4.4.1 Spin-weighted parabolic harmonics
4.4.2 Spin-weighted elliptic harmonics
4.5 Applications
4.5.1 Solution of the vector Helmholtz equation
4.5.2 Divergenceless vector fields
4.5.3 Solution of the Dirac equation
5 Spinor Algebra
5.1 The spinor equivalent of a tensor
5.2 The orthogonal and spin groups
5.2.1 Positive definite metric
5.2.2 Indefinite metric
5.3 Algebraic classification
5.4 The triad defined by a spinor
6 Spinor Analysis
6.1 Covariant defferentiation
6.2 Curvature
6.3 Spin weight and priming operation
6.3.1 Positive definite metric
6.3.2 Indefinite metric
6.4 Metric connections with torsion
6.5 Congruences of curves
6.6 Applications
7 Applications to General Relativity
7.1 Spacelike hypersurfaces
7.2 Timelike hypersurfaces
7.3 Stationary space-times
Appendix: Spinors in the Four-Dimensional Space-Time
References
Index
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