Null Killing vectors and geometry of null strings in Einstein spaces

TL;DR

This paper explores Einstein spaces with null Killing vectors, analyzing their geometric structures called null strings, and identifies specific complex metrics with these properties, including their Lorentzian and ultrahyperbolic slices.

Contribution

It characterizes null Killing vectors in Einstein spaces, links them to null strings and twistor surfaces, and explicitly constructs complex metrics with these features.

Findings

Spaces are hyperheavenly or heavenly spaces.

Explicit complex metrics with null Killing vectors are derived.

Lorentzian and ultrahyperbolic slices of these metrics are discussed.

Abstract

Einstein complex spacetimes admitting null Killing or null homothetic Killing vectors are studied. These vectors define totally null and geodesic 2-surfaces called the null strings or twistor surfaces. Geometric properties of these null strings are discussed. It is shown, that spaces considered are hyperheavenly spaces (HH-spaces) or, if one of the parts of the Weyl tensor vanishes, heavenly spaces (H-spaces). The explicit complex metrics admitting null Killing vectors are found. Some Lorentzian and ultrahyperbolic slices of these metrics are discussed.