Classification of complex and real, vacuum spaces of the type $[\textrm{N}] \otimes [\textrm{N}]$

TL;DR

This paper classifies complex and real vacuum spaces with Weyl tensor parts of type [N], based on properties of null congruences and their intersections, identifying six distinct types and presenting new examples.

Contribution

It introduces a classification scheme for $[ extrm{N}] imes [ extrm{N}]$ spaces based on null congruences and intersection properties, revealing six types and providing new metric examples.

Findings

Six distinct types of $[ extrm{N}] imes [ extrm{N}]$ spaces identified.

New Lorentzian slice examples of complex metrics presented.

Some $[ extrm{N}] imes [ extrm{N}]$ spaces lack Lorentzian slices.

Abstract

Complex and real, vacuum spaces with both self-dual and anti-self-dual parts of the Weyl tensor being of the type [N] are considered. Such spaces are classified according to two criteria. The first one takes into account the properties of the congruences of totally null, geodesic 2-dimensional surfaces (the null strings). The second criterion is based on investigations of the properties of the intersection of these congruences. It is proved that there exist six distinct types of the $[\textrm{N}] \otimes [\textrm{N}]$ spaces. New examples of the Lorentzian slices of the complex metrics are presented. Also, some type $[\textrm{N}] \otimes [\textrm{N}]$ spaces which do not posses Lorentzian slices are considered.