Introduction to 2-Spinors in General Relativity

The following sections are included:

• Preface

• Contents

The following sections are included:

• Minkowski space

• The null cone and Riemann sphere

• Spin transformations and spin matrices

• Flagpoles and flag planes

• Spin-space

• Exercises

The following sections are included:

• Abstract index notation

• Complex conjugation of spinor components

• Vector bases and abstract indices

• Levi–Civita spinor

• Spinor dyad basis and its components

• Spinor symmetry operations

• The connection between world-tensors and spinors
  • Infeld–van der Waerden symbols
  • Null vectors

• The decomposition of spinors
  • The spinor equivalent of a symmetric valence two tensor
  • The spinor equivalent of the electromagnetic field tensor

• The canonical decomposition of symmetric spinors
  • Classification of the electromagnetic spinor
  • Classification of the Weyl spinor

• Exercises

The following sections are included:

• Spinor form of the covariant derivative

• The curvature spinors
  • Symmetries of the curvature spinors
  • The Ricci spinor
  • The Weyl spinor

• Spinor equivalent of the Ricci identities

• Spinor equivalent of the Bianchi identities

• The Newman–Penrose spin coefficient formalism
  • Spin coefficients
  • The Newman–Penrose field equations

• Newman–Penrose quantities under Lorentz transformations
  • Null rotation with l fixed
  • Null rotation with n fixed
  • Lorentz spin-boost transformation

• Miscellaneous transformations
  • l ↔ n and
  • Prime operation
  • Asterisk operation

• Geroch–Held–Penrose formalism
  • Spin- and boost-weighted scalars
  • Weighted and unweighted spin coefficients
  • Weighted differential operators
  • The Geroch–Held–Penrose field equations
  • A brief note on the modified GHP formalism

• Goldberg–Sachs theorem

• Exercises

The following sections are included:

• Introduction

• Lanczos' Lagrangian

• Lanczos' gauge conditions

• The Lanczos spinor

• The spinor version of the Weyl–Lanczos equations

• The Lanczos coefficients

• The Weyl–Lanczos equations in spin coefficient form

• The Ricci–Lanczos equations in spin coefficient form

• The behaviour of Lanczos coefficients under Lorentz transformations
  • Null rotation with l fixed
  • Null rotation with n fixed
  • Spin-boost transformation

• Miscellaneous transformations
  • l ↔ n and
  • Prime operation
  • Asterisk operation

• The Weyl–Lanczos equations in GHP form

• Solutions of the Weyl–Lanczos equations
  • Petrov type O space-times
  • Petrov type N space-times
  • Petrov type III space-times
  • Lanczos coefficients for the Schwarzschild metric

• A brief note on the Lanczos spinor/tensor

• Exercises

The following sections are included:

• Appendix A Aspects of general relativity
  • The space-time of general relativity
    • Space-time as a differentiable manifold
    • The tangent space T_Po(M) at a point P_O of a manifold M
    • Tensor algebra
    • Tensor detection
    • An illustrative example from classical mechanics
    • Tensor fields
    • An illustrative example from special relativity
  • Riemannian geometry and tensor analysis
    • Introduction
    • Covariant derivatives
    • The curvature tensor
    • The Bianchi identities
    • Riemannian geometry and geodesics
  • General relativity
    • The field equations
    • The Newtonian approximation
    • The Schwarzschild solution
    • Planetary orbits
    • The deflexion of light
    • The gravitational red shift
  • Exercises

• Bibliography

• Index