The Gravitational Field Equations of a Twisted Skyrmion String: Numerical Solution

TL;DR

This paper investigates the gravitational effects of a twisted Skyrmion string by numerically solving the coupled Einstein field equations using MATLAB, revealing how twist influences the spacetime structure of such topological defects.

Contribution

It introduces a numerical approach to solve the Einstein equations for a twisted Skyrmion string, incorporating twist effects into the gravitational field analysis.

Findings

Numerical solutions of Einstein equations for twisted Skyrmion strings.

Impact of twist on the spacetime geometry around the string.

Demonstration of the use of MATLAB's ode45 for complex gravitational systems.

Abstract

We study nonlinear sigma model, especially Skyrme model with twist: twisted Skyrmion string where twist term, $mkz$, is indicated in vortex solution. To add gravity, we replace $\eta^{\mu\nu}$ in Lagrangian system with a space-time metric tensor, $g^{\mu\nu}$, which in view of the time-independence and cylindrical symmetry of the assumed vortex solution is taken to be a function of $r$ alone. We use ode45 for numerical calculation, i.e. a tool box in Matlab to solve coupled Einstein field equations which have ordinary differential equations (ODE) form.