## An Introduction to Twistor TheoryThis book is an introduction to twistor theory and modern geometrical approaches to space-time structure at the graduate or advanced undergraduate level. It will be valuable also to the physicist as an introduction to some of the mathematics that has proved useful in these areas, and to the mathematician as an example of where sheaf cohomology and complex manifold theory can be used in physics. |

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### Contents

Introduction | 1 |

Review of Tensor Algebra and Calculus | 5 |

Lorentzian Spinors at a Point | 11 |

Spinor Fields | 25 |

Compactified Minkowski Space | 33 |

The Geometry of Null Congruences | 45 |

The Geometry of Twistor Space | 53 |

Solving the Zero Rest Mass Equations I | 65 |

Solving the Zero Rest Mass Equations II | 91 |

The Twisted Photon and YangMills Constructions | 99 |

The NonLinear Graviton | 105 |

Penroses QuasiLocal Momentum and Angular Momentum | 119 |

Functionals on Zero Rest Mass Fields | 137 |

Further Developments and Conclusions | 147 |

The GHP Equations | 163 |

Sheaf Cohomology and Free Fields | 71 |

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### Common terms and phrases

a-plane analytic Bianchi identity bivector bundle calculate chapter choice cohomology complex complex manifold components condition conformal congruence connection consider constant construction contour coordinates corresponding cover curvature curves define definition deformation derivative describe determines diagram discussion elements equations equivalent exact sequence example exercise fact field figure Finally flat follows function Further geometrical geometry give given holomorphic homogeneous of degree identity indices infinity integral intersection introduce mass means metric Minkowski space momentum Note null cone null geodesics null vector obtained original pair particular Penrose plane primed projective relative represented respectively result rotation satisfying sheaf Show simple solutions solve space-time sphere spinor structure suppose surfaces symmetric tangent tensor Theorem transformation twistor twistor space twistor theory two-surface vanishes vector field Ward Weyl written zero