We derive the equations of motion from the Einstein-Skyrme action, incorporating spherical symmetry, polar/areal slicing and the so-called "hedgehog" ansatz for the SU(2) Skyrme field. Skyrmions, and their topological characteristics, are briefly reviewed. The solutions can be classified by their winding number, called B for "baryon number", due to Skyrme's original intention of modelling baryons as stable configurations of this particular SU(2) gauge theory. The baryon number in this model is the difference of the Skyrme field at the origin and at infinity, divided by pi. We concentrate on the B=1 sector at this stage.
The static solution found by Droz, Heusler and Straumann (1991) is recovered. It is found by a shooting procedure, which turns out to be sensitive to the coupling constant kappa. For kappa higher than a critical value of 0.0202, the shooting consistently arrives below the desired field value (pi) at infinity and never converges. Thus, like in the literature, we find no static solutions for a coupling constant higher than 0.0202. On the other hand, for kappa going to zero, there are solutions, which carry a finite mass 72.9, the mass of the flat-space skyrmion.
We have an RNPL code running for the dynamical evolution. A tanh((r-r0)/delta) initial data-pulse is used. We observe mass conservation and quadratic convergence away from the origin, but we notice problems close to the origin, especially in the geometrical variables such as grr. We recreate the evolution of the field "flopping onto" the static solution.
We intend to first of all cure our origin problems and ensure regularity. Later work includes verifying Type I behavior, studying the B=0 sector (Gaussian initial data) and hopefully Type II critical behavior. Eventually we hope to have our adaptive mesh refinement code running.