Algebraically Special, Real Alpha Geometries

TL;DR

This paper investigates four-dimensional neutral geometries with algebraically special self-dual Weyl curvature, using spinor methods and conformal rescalings to derive key equations and describe their local structure.

Contribution

It introduces a spinor-based approach to describe algebraically special real alpha-geometries and derives the hyperheavenly equation from conformal rescaling formulas.

Findings

Behavior of Walker geometry under conformal rescalings

Derivation of the hyperheavenly equation

Characterization of algebraically degenerate self-dual Weyl curvature

Abstract

We exploit the spinor description of four-dimensional Walker geometry, and conformal rescalings of such, to describe the local geometry of four-dimensional neutral geometries with algebraically degenerate self-dual Weyl curvature and an integrable distribution of alpha-planes (algebraically special real alpha-geometry). In particular, we determine the behaviour of (four-dimensional neutral) Walker geometry under conformal rescalings and provide a derivation of the hyperheavenly equation from conformal rescaling formulae.