Nontrivial causal structures engendered by knotted solitons

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 π 3 (S 2 ). As a consequence, it follows that the preimage curves in 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.

From Hopf fibrations to exotic causal replacements

“…Using the effective metric technique (see, for instance, [24]), we work out numerically several interesting situations and discuss how the topological charge may affect the behaviour of rays. In this sense, the present work is a natural generalization of previous results sketched in [25].…”

Characteristic analysis for integrable soliton models on two-dimensional target spaces

“…the Dirichlet term should enter as a rather small addition. Formally, this choice is equivalent to the Faddeev-Niemi model with a potential and admits nice characteristics for sufficiently slowly varying background solutions [27]. However, this model is not integrable and it can be considerably difficult to obtain…”

“…For well-posedness to hold, it is crucial that small disturbances (or, equivalently, discontinuities) in the initial data propagate in a controlled/predictable way with a definite finite velocity of propagation. Roughly speaking, this means that the associated characteristic surfaces about a background solution have to be described by algebraic varieties with the topology of convex cones for each point of M 1+3 (see [26] and [27] for a similar analysis in the contexts of the Baby-Skyrme and Faddeev models ).…”