Inspecting baby Skyrmions with effective metrics

In the present paper we investigate the causal structure of the baby Skyrme model using appropriate geometrical tools. We discuss several features of excitations propagating on top of background solutions and show that the evolution of high frequency waves is governed by a curved effective geometry. Examples are given for which the effective metric describes the interaction between waves and solitonic solutions such as kinks, antikinks, and hedgehogs. In particular, it is shown how violent processes involving the collisions of solitons and antisolitons may induce metrics which are not globally hyperbolic. We argue that it might be illuminating to calculate the effective metric as a diagnostic test for pathological regimes in numerical simulations.

Figure 1

Causal structure of the kink (antikink) solution in the $t$-$x$ plane. The colors represent the behavior of the function $\sqrt{\tilde{h}}$ with parameters ${\kappa }^2=1$ and $\mu =1$. In this case it is well defined for all possible spacetime events. Note, however, that it only varies significantly near the kink (antikink), yielding the usual cone for distant excitations.

Figure 2

Color diagrams representing the function $\sqrt{\tilde{h}}$ for the kink-kink collision in the $t$-$x$ plane for $0.1\le c\le 0.9$. Note that the function is real for all values of the parameter.

Figure 3

Causal structure associated with the kink-kink collision in the $t$-$x$ plane for $c=0.7$. Note that the velocities of propagation are drastically modified near the collision.

Figure 4

Causal structure of the kink-antikink collision in the $t$-$x$ plane for $c=0.5$. The colors represent the function $\sqrt{\tilde{h}}$. Note that there exists a region where it vanishes. The white hole in the middle of the figure represents a region with an elliptic signature.

Figure 5

Radial velocity in terms of $r$ for the first two baby Skyrmions $n=1$ (blue curve) and $n=2$ (red curve). Note that for $r\to \infty$ and $r=0$ the velocities coincide with the velocity of light.