Introduction to Complex Space-Time

Abstract

This chapter begins by describing the physical and mathematical motivations for studying complex space-times or real Riemannian four-manifolds in gravitational physics. 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 paracompact Hausdorff spaces where local coordinates transform by complex-analytic transformations. Examples are given such as complex projective space P _m non-singular sub-manifolds of P _m , and orientable surfaces. The plan of the whole monograph is finally presented, with emphasis on two-component spinor calculus, Penrose transform and Penrose formalism for spin-3/2 potentials.