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

Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology

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11 pages, a plain TeX file, submitted to ``Classical and Quantum Gravity''

Scientific paper

10.1088/0264-9381/14/2/021

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.

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