Nontrivial Causal Structures Engendered by Knotted Solitons

TL;DR

This paper demonstrates that string-like solutions in the Fadeev-Niemi model create a unique causal structure described by an effective metric, which depends on the soliton's energy-momentum tensor and topological invariants, impacting the understanding of dynamic solutions.

Contribution

It reveals that the causal structure of knotted solitons in the FN model is governed by an effective metric linked to topological invariants, a novel insight into their geometry.

Findings

Causal surfaces depend on the energy-momentum tensor.

Pre-image curves define invariant directions for cones.

Results are relevant for analyzing soliton collisions.

Abstract

It is shown that the causal structure associated to string-like solutions of the Fadeev-Niemi (FN) model is described by an effective metric. Remarkably, the surfaces characterising the causal replacement depend on the energy momentum tensor of the background soliton and carry implicitly a topological invariant $\pi_{3}(\mathbb{S}^2)$. As a consequence, it follows that the pre- image curves in $\mathbb{R}^3$ nontrivialy define directions where the cones remain unchanged. It turns out that these results may be of importance in understanding time dependent solutions (collisions/scatterings) numerically or analytically.