3-D Spinors Spin-Weighted Functions and their Applications

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. 4 Parabolic and elliptic coordinates. - 4. 5 Applications. - 5 Spinor Algebra. - 5. 1 The spinor equivalent of a tensor. - 5. 2 The orthogonal and spin groups. - 5. 3 Algebraic classification. - 5. 4 The triad defined by a spinor. - 6 Spinor Analysis. - 6. 1 Covariant differentiation. - 6. 2 Curvature. - 6. 3 Spin weight and priming operation. - 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. Language: English