This book is the first to present a comprehensive development of space-time geometry using the 2-spinor formalism. There are also several other new features in our presentation. One of these is the systematic and consistent use of the abstract index approach to tensor and spinor calculus. We hope that the purist differential geometer who casually leafs through the book will not automatically be put off by the appearance of numerous indices. Except for the occasional bold-face upright ones, our indices differ from the more usual ones in being abstract markers without reference to any basis or coordinate system. Our use of abstract indices leads to a number of simplifications over conventional treatments.

Preface

 1 The geometry of world-vectors and spin-vectors

1.1 M inkowski vector space

1.2 Null directions and spin transformations

1.3 Some properties of Lorentz transformations

1.4 Null flags and spin-vectors

1.5 Spinorial objects and spin structure

1.6 The geometry ofspinor operations

 2 Abstract indices and spinor algebra

2.1 Motivation for abstract-index approach

2.2 The abstract-index formalism for tensor algebra

2.3 Bases

2.4 The total reflexivity of on a manifold

2.5 Spinor algebra

 3 Spinors and worid-tensors

3.1 World-tensors as spinors

3.2 Null flags and complex null vectors

3.3 Symmetry operations

3.4 Tensor representation of spinor operations

3.5 Simple propositions about tensors and spinors at a point

3.6 Lorentz transformations

 4 Differentiation and curvature

4.1 Manifolds

4.2 Covariant derivative

4.3 Connection-independent derivatives

4.4 Differentiation ofspinors

4.5 Differentiation ofspinor components

4.6 The curvature spinors

4.7 Spinor formulation of the Einstein-Cartan-Sciama-Kibble theory

4.8 The Weyl tensor and the BeI-Robinson tensor

4.9 Spinor form of commutators

4.10 Spinor form of the Bianchi identity

4.11 Curvature spinors and spin-coefficients

4.12 Compacted spin-coefficient formalism

4.13 Cartan's method

4.14 Applications to 2-surfaces

4.15 Spin-weighted spherical harmonics

 5 Fields in space-time

5.1 The electromagnetic field and its derivative operator

5.2 Einstein-Maxwell equations in spinor form

5.3 The Rainich conditions

5.4 Vector bundles

5.5 Yang-Mills fields

5.6 Conformal rescalings

5.7 Massless fields

5.8 Consistency conditions

5.9 Conformal invariance of various field quantities

5.10 Exact sets of fields

5.11 Initial data on a light cone

5.12 Explicit field integrals

Appendix: diagrammatic notation

References

Subject and author index

Index of symbols