Universality in the critical collapse of the Einstein-Maxwell system

We report on critical phenomena in the gravitational collapse of the electromagnetic field in axisymmetry using cylindrical coordinates. We perform detailed numerical simulations of four families of dipole and quadrupole initial data fine-tuned to the onset of black hole formation. It has been previously observed that families that bifurcate into two on-axis critical solutions exhibit distinct growth characteristics from those that collapse at the center of symmetry. In contrast, our results indicate similar growth characteristics and periodicity across all families of initial data, including those examined in earlier works. More precisely, for all families investigated, we observe power-law scaling for the maximum of the electromagnetic field invariant ($\mathrm{max}|{F}_{\ensuremath{\mu}\ensuremath{\nu}}{F}^{\ensuremath{\mu}\ensuremath{\nu}}|\ensuremath{\sim}|p\ensuremath{-}{p}^{\ensuremath{\star}}{|}^{\ensuremath{-}2\ensuremath{\gamma}}$) with $\ensuremath{\gamma}\ensuremath{\approx}0.149(9)$. We find evidence of approximate discrete self-similarity in near-critical time evolutions with a log-scale echoing period of $\mathrm{\ensuremath{\Delta}}\ensuremath{\approx}0.62(8)$ across all families of initial data. Our methodology, while reproducing the results of prior studies up to a point, provides new insights into the later stages of critical searches and we propose a mechanism to explain the observed differences between our work and the previous calculations.

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