Non-Twisting and Twisting Solutions of the Einstein Field Equations of a Skyrmionic String

TL;DR

This paper investigates solutions to Einstein's field equations coupled with a Skyrme model, demonstrating the non-existence of certain infinite and finite radius solutions with or without twist.

Contribution

It proves the non-existence of both non-twisting and twisting solutions extending from zero to infinity, and also shows no finite radius solutions satisfy boundary junction conditions.

Findings

No solutions extend from r=0 to r=∞ for both twists.

Finite radius solutions cannot satisfy junction conditions.

Twisting solutions do not exist under the studied conditions.

Abstract

We construct non-linear sigma model plus Skyrme term (Skyrme model) with a twist in the gravitational field. We try to solve the Einstein field equations for small and large values of $r$, with and without twist. We prove that no non-twisting or twisting solutions extending from $r=0$ to $r=\infty$ exist. At last, we try to solve non-twisting and twisting solutions of the Einstein field equations with a finite radius. We find that there are no solutions with a finite radius that can satisfy the junction conditions at the boundary radius $r=r_b$, where $r_b$ is a finite radius.