Vaccum solutions of five-dimensional Einstein equations generated by inverse scattering method II : Production of black ring solution

TL;DR

This paper demonstrates how to generate five-dimensional vacuum solutions, including the black ring, using the inverse scattering method, by applying successive solitonic transformations to known seed metrics.

Contribution

It reproduces the black ring solution via inverse scattering transformations starting from the Euclidean Levi-Civita metric, illustrating a method to find new higher-dimensional solutions.

Findings

Successfully reproduces the black ring solution.

Shows a systematic process for generating solutions via solitonic transformations.

Provides a framework for discovering new exact solutions in higher-dimensional gravity.

Abstract

We study vacuum solutions of five-dimensional Einstein equations generated by the inverse scattering method. We reproduce the black ring solution which was found by Emparan and Reall by taking the Euclidean Levi-Civita metric plus one-dimensional flat space as a seed. This transformation consists of two successive processes; the first step is to perform the three-solitonic transformation of the Euclidean Levi-Civita metric with one-dimensional flat space as a seed. The resulting metric is the Euclidean C-metric with extra one-dimensional flat space. The second is to perform the two-solitonic transformation by taking it as a new seed. Our result may serve as a stepping stone to find new exact solutions in higher dimensions.