Static Solution of the General Relativistic Nonlinear $\sigma$-Model Equation

TL;DR

This paper analyzes static solutions of the nonlinear sigma-model coupled with gravity, revealing scale invariance and a specific geometric structure at large distances, contributing to understanding hadron models in curved spacetime.

Contribution

It provides a detailed analysis of static solutions in the gravitational nonlinear sigma-model, highlighting their scale invariance and geometric properties at large distances.

Findings

No scale parameter determines the size of solutions.

Winding number of solutions is 1/2.

Spatial geometry resembles flat space with a missing solid angle.

Abstract

The nonlinear $\sigma$-model is considered to be useful in describing hadrons (Skyrmions) in low energy hadron physics and the approximate behavior of the global texture. Here we investigate the properties of the static solution of the nonlinear $\sigma$-model equation coupled with gravity. As in the case where gravity is ignored, there is still no scale parameter that determines the size of the static solution and the winding number of the solution is $1/2$. The geometry of the spatial hyperspace in the asymptotic region of large $r$ is explicitly shown to be that of a flat space with some missing solid angle.