Regular Type III and Type N Approximate Solutions

TL;DR

This paper introduces new regular Type III and Type N approximate solutions in general relativity, utilizing complex transformations on Robinson-Trautman metrics to ensure regularity in the linear approximation.

Contribution

It presents a novel method using complex transformations to find regular approximate solutions of Type III and N, differing from classical approaches.

Findings

Existence of regular Type III and N solutions in the linear approximation.

Use of complex transformations on Robinson-Trautman metrics.

Regularity criterion based on scalar boundedness and decay at infinity.

Abstract

New type III and type N approximate solutions which are regular in the linear approximation are shown to exist. For that, we use complex transformations on self-dual Robinson-Trautman metrics rather then the classical approach. The regularity criterion is the boundedness and vanishing at infinity of a scalar obtained by saturating the Bel-Robinson tensor of the first approximation by a time-like vector which is constant with respect to the zeroth approximation.