February 1, 2008 20:37 WSPC/INSTRUCTION FILE esposito International Journal of Geometric Methods in Modern Physics c World Scientific Publishing Company FROM SPINOR GEOMETRY TO COMPLEX GENERAL RELATIVITY∗ GIAMPIERO ESPOSITO INFN, Sezione di Napoli, and Dipartimento di Scienze Fisiche, Universit`aFederico II, Complesso Universitario di Monte S. Angelo, Via Cintia, Edificio N’ 80126 Napoli, Italy
[email protected] Received (1 April 2005) An attempt is made of giving a self-contained introduction to holomorphic ideas in gen- eral relativity, following work over the last thirty years by several authors. The main top- ics are complex manifolds, two-component spinor calculus, conformal gravity, α-planes in Minkowski space-time, α-surfaces and twistor geometry, anti-self-dual space-times and Penrose transform, spin-3/2 potentials, heaven spaces and heavenly equations. Keywords: Two-component spinors; twistors; Penrose transform. 1. Introduction to Complex Space-Time The physical and mathematical motivations for studying complex space-times or arXiv:hep-th/0504089v2 23 May 2005 real Riemannian four-manifolds in gravitational physics are first described. They originate from algebraic geometry, Euclidean quantum field theory, the path- integral approach to quantum gravity, and the theory of conformal gravity. The theory of complex manifolds is then briefly outlined. Here, one deals with para- compact Hausdorff spaces where local coordinates transform by complex-analytic transformations. Examples are given such as complex projective space Pm, non- singular sub-manifolds of Pm, and orientable surfaces. The plan of the whole paper is eventually presented, with emphasis on two-component spinor calculus, Penrose 3 transform and Penrose formalism for spin- 2 potentials.