Spin coefficients for four-dimensional neutral metrics, and null geometry

TL;DR

This paper introduces spin coefficients for four-dimensional neutral metrics, explores their interpretation via null geometry, and extends Walker geometry and Plebański's heavenly equation to this context.

Contribution

It develops a notation for spin coefficients in neutral signature metrics and generalizes Plebański's heavenly equation within Walker geometries.

Findings

Introduced spin coefficient notation for neutral metrics

Connected spin coefficients to null geometry themes

Extended Plebański's heavenly equation to Walker geometries

Abstract

Notation for spin coefficients for metrics of neutral signature in four dimensions is introduced. The utility and interpretation of spin coefficients is explored through themes in null geometry familiar from (complex) general relativity. Four-dimensional Walker geometry is exploited to provide examples and the generalization of the real neutral version of Pleba\~nski's (1975) second heavenly equation to certain Walker geometries given in Law and Matsushita [16] is extended further.