Critical Phenomena in the Gravitational Collapse of Electromagnetic Waves

We numerically investigate the threshold of black-hole formation in the gravitational collapse of electromagnetic waves in axisymmetry. We find approximate power-law scaling ${\ensuremath{\rho}}_{\mathrm{max}}\ensuremath{\sim}({\ensuremath{\eta}}_{*}\ensuremath{-}\ensuremath{\eta}{)}^{\ensuremath{-}2\ensuremath{\gamma}}$ of the maximum density in the time evolution of near-subcritical data with $\ensuremath{\gamma}\ensuremath{\simeq}0.145$, where $\ensuremath{\eta}$ is the amplitude of the initial data. We directly observe approximate discrete self-similarity in near-critical time evolutions with a log-scale echoing period of $\mathrm{\ensuremath{\Delta}}\ensuremath{\simeq}0.55$. The critical solution is approximately the same for two families of initial data, providing some evidence of universality. Neither the discrete self-similarity nor the universality, however, are exact. We speculate that the absence of an exactly discrete self-similarity might be caused by the interplay of electromagnetic and gravitational wave degrees of freedom, or by the presence of higher-order angular multipoles, or both, and discuss implications of our findings for the critical collapse of vacuum gravitational waves.