Vacuum Solutions of Einstein's Equations in Parabolic Coordinates

TL;DR

This paper introduces a straightforward method for deriving vacuum solutions of Einstein's equations in parabolic coordinates from cylindrical symmetric solutions, and explores its generalizations and connections to soliton solutions.

Contribution

It presents a novel method to generate vacuum solutions in parabolic coordinates and discusses its relation to existing soliton-generating techniques.

Findings

Method successfully derives solutions in parabolic coordinates.

Generalization extends applicability to broader cases.

Links established between the method and Belinsky-Zakharov solutions.

Abstract

We present a simple method to obtain vacuum solutions of Einstein's equations in parabolic coordinates starting from ones with cylindrical symmetries. Furthermore, a generalization of the method to a more general situation is given together with a discussion of the possible relations between our method and the Belinsky-Zakharov soliton-generating solutions.