From Spinor Geometry to Complex General Relativity

TL;DR

This paper introduces holomorphic methods in general relativity, covering complex manifolds, spinor calculus, twistor theory, and anti-self-dual space-times, highlighting their mathematical structures and physical implications.

Contribution

It provides a self-contained overview of complex geometric techniques and their applications in understanding various aspects of general relativity.

Findings

Connection between twistor geometry and anti-self-dual solutions

Representation of space-times using spinor calculus

Insights into heavenly equations and heaven spaces

Abstract

An attempt is made of giving a self-contained (although incomplete) introduction to holomorphic ideas in general relativity, following work over the last thirty years by several authors. The main topics are complex manifolds, two-component spinor calculus, conformal gravity, alpha-planes in Minkowski space-time, alpha-surfaces and twistor geometry, anti-self-dual space-times and Penrose transform, spin-3/2 potentials, heaven spaces and heavenly equations.