Some Higher Dimensional Vacuum Solutions

TL;DR

This paper develops a method to generate higher-dimensional Ricci flat solutions from four-dimensional Ricci flat metrics, including colliding gravitational wave spacetimes, and analyzes their singularities.

Contribution

It introduces a procedure to construct higher-dimensional vacuum solutions from four-dimensional geometries, extending the understanding of Einstein's equations in higher dimensions.

Findings

Explicit higher-dimensional Szekeres metrics provided

Method applied to colliding gravitational vacuum spacetimes

Singularity behaviors of solutions analyzed

Abstract

We study an even dimensional manifold with a pseudo-Riemannian metric with arbitrary signature and arbitrary dimensions. We consider the Ricci flat equations and present a procedure to construct solutions to some higher (even) dimensional Ricci flat field equations from the four diemnsional Ricci flat metrics. When the four dimensional Ricci flat geometry correponds to a colliding gravitational vacuum spacetime our approach provides an exact solution to the vacuum Einstein field equations for colliding graviational plane waves in an (arbitrary) even dimensional spacetime. We give explicitly higher dimensional Szekeres metrics and study their singularity behaviors.