Non-vacuum twisting type N metrics

TL;DR

This paper derives a simplified system of equations for twisting type N Einstein metrics, explores conformally related metrics with symmetries, and provides an example of a non-Einstein metric satisfying Bach's equations.

Contribution

It introduces a maximally reduced system for twisting type N metrics and presents new examples within the Feferman class, expanding understanding of their geometric properties.

Findings

Reduced equations for twisting type N Einstein metrics.

Classification of conformally equivalent metrics with symmetries.

Example of a non-Einstein metric satisfying Bach's equations.

Abstract

A maximally reduced system of equations corresponding to the twisting type N Einstein metrics is given. When the cosmological constant $\lambda\to 0$ they reduce to the standard equations for the vacuum twisting type N's. All the metrics which are conformally equivalent to the twisting type N metrics and which admit 3-dimensional conformal group of symmetries are presented. In the Feferman class of metrics an example is given of a twisting type N metric which satisfies Bach's equations but is not Einstein.