An Iterative Approach to Twisting and Diverging, Type N, Vacuum Einstein Equations: A (Third-Order) Resolution of Stephani's `Paradox'

TL;DR

This paper resolves a long-standing paradox in vacuum Einstein equations by demonstrating that third-order perturbative solutions can be regular, challenging previous claims that such solutions are inherently singular and cannot model gravitational waves.

Contribution

The authors introduce a third-order iterative method to find regular solutions to twisting, type-N vacuum Einstein equations, overturning prior assumptions about their singularity.

Findings

Third-order perturbations yield regular solutions

Solutions become flat before losing twisting

Challenges previous claims about solution singularities

Abstract

In 1993, a proof was published, within ``Classical and Quantum Gravity,'' that there are no regular solutions to the {\it linearized} version of the twisting, type-N, vacuum solutions of the Einstein field equations. While this proof is certainly correct, we show that the conclusions drawn from that fact were unwarranted, namely that this irregularity caused such solutions not to be able to truly describe pure gravitational waves. In this article, we resolve the paradox---since such first-order solutions must always have singular lines in space for all sufficiently large values of $r$---by showing that if we perturbatively iterate the solution up to the third order in small quantities, there are acceptable regular solutions. That these solutions become flat before they become non-twisting tells us something interesting concerning the general behavior of solutions describing gravitational radiation from a bounded source.