STABILITY ANALYSIS OF SELF-GRAVITATING SKYRMIONS

We investigate the stability of our recently constructed self-gravitating skyrmions in the framework of small time-dependent perturbations. It is shown that the frequency spectrum for radial perturbations of the coupled Einstein-Skyrme equations can be reduced to the energy spectrum of a p-wave Schrodinger equation with a bounded effective potential, which is determined by the equilibrium solution. Bound states of this Schrodinger equation correspond to exponentially growing modes. It turns out that there are no such modes, as long as the relevant dimensionless coupling constant (kappa) is less than a critical value, kappa(c). For larger values of kappa there is exactly one unstable mode. Therefore, the Skyrme "stars" become unstable (in the sense of Liapunov) for kappa > kappa(c), in spite of the fact that their topological winding number is conserved. The linear stability for kappa < kappa(c) (with respect to radial modes) is, of course, only a necessary condition for (non-linear) stability. Similar results are expected to hold for our new black hole solutions of the Einstein-Skyrme system.