On the local structure of Lorentzian Einstein manifolds with parallel distribution of null lines

TL;DR

This paper investigates the local geometric structure of Lorentzian Einstein manifolds with a parallel null line distribution, simplifying the Einstein equations via coordinate transformations to relate them to Einstein Riemannian metrics.

Contribution

It introduces a method to simplify Walker coordinates on such manifolds, reducing Einstein equations to a family of Einstein Riemannian metrics.

Findings

Walker coordinates can be simplified on Lorentzian Einstein manifolds with null lines.

The Einstein equations reduce to equations on Einstein Riemannian metrics.

The approach clarifies the geometric structure of these manifolds.

Abstract

We study transformations of coordinates on a Lorentzian Einstein manifold with a parallel distribution of null lines and show that the general Walker coordinates can be simplified. In these coordinates, the full Lorentzian Einstein equation is reduced to equations on a family of Einstein Riemannian metrics.