Topological Properties from Einstein's Equations?

TL;DR

This paper introduces a novel method to analyze the global topological properties of space-times by using Einstein's equations and immersion techniques, revealing different topologies in static, spherically symmetric vacuum solutions.

Contribution

It presents a new procedure combining immersion theory and Einstein's equations to extract topological information of space-times, implemented through an algebraic computing algorithm.

Findings

Space-times with different topologies were obtained.

The method successfully applies to static, spherically symmetric vacuum solutions.

A computational algorithm was developed for this analysis.

Abstract

In this work we propose a new procedure for to extract global information of a space-time. We considered a space-time immersed in a higher dimensional space and we formulate the equations of Einstein through of the Frobenius conditions to immersion. Through of an algorithm and the implementation into algebraic computing system we calculate normal vectors from the immersion to find out the second fundamental form. We make a application for space-time with spherical symmetry and static. We solve the equations of Einstein to the vacuum and we obtain space-times with different topologies.