## 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 arid Conclusions | 147 |

The GHP Equations | 163 |

Sheaf Cohomology and Free Fields | 71 |

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### Common terms and phrases

a-plane Baston bivector chapter coboundary cocycle cohomology groups compactified Minkowski space complex conjugate complex manifold conformal group conformal metric congruence consider constant construction contour integral coordinates corresponding curvature curves define definition deformation Dolbeault Dolbeault cohomology dyad equivalent example exercise fibre field equations flat space function elements geometry given Hermitian homogeneous of degree Hughston infinity intersection isomorphism Lie derivative line bundle linear linearised long exact sequence Lorentz matrix Maxwell field Minkowski space momentum null cone null geodesic null vector Penrose plane quasi-local mass represented Riemann rotation satisfying scalar sections sheaf cohomology short exact sequence Show solutions space-time sphere spinor field surface symmetric tangent tetrad Theorem time-like topological transformation twistor diagrams twistor equation twistor function twistor space twistor theory two-surface twistor vanishes vector field vector space Ward Weyl tensor Yang-Mills z.r.m. field zero rest mass