Critical phenomena in the gravitational collapse of electromagnetic dipole and quadrupole waves

We report on critical phenomena in the gravitational collapse of electromagnetic waves. Generalizing earlier results that focused on dipole electromagnetic waves, we compare here with quadrupole waves in axisymmetry. We perform numerical simulations of dipole and quadrupole wave initial data, fine tuning both sets of data to the onset of black hole formation in order to study the critical solution and related critical phenomena. We observe that different multipole moments have different symmetries, indicating that the critical solution for electromagnetic waves cannot be unique, at least not globally. This is confirmed in our numerical simulations: while dipole data lead to a single center of collapse, at the center of symmetry, quadrupole data feature two separate centers of collapse on the symmetry axis, above and below the center of symmetry-reminiscent of similar findings reported for critical collapse of vacuum gravitational waves. While the critical solution for neither the dipole nor the quadrupole data is exactly self-similar, we find that their approximate echoing periods appear to differ, as do the critical exponents. We discuss whether the centers of collapse found for dipole and quadrupole data might all have the same properties, which would suggest a "local uniqueness" of the critical solution. Instead, we provide some evidence-including the differing echoing periods and critical exponents-suggesting that the critical solutions are distinct even locally. We speculate on the implications of our findings for critical phenomena in the collapse of vacuum gravitational waves, which share with electromagnetic waves the absence of a spherically symmetric critical solution.