  In this study we show that, from arbitrarily dispersed initial data, both the
concentration of electromagnetic fields and the focusing of gravitational waves
could lead to the formation of trapped surfaces. We establish a scale-critical
semi-global existence result from the past null infinity for the
Einstein-Maxwell system by assigning the signature for decay rates to geometric
quantities and Maxwell components. This result generalizes an approach of the
first author on studying Einstein vacuum equations by employing additional
elliptic estimates and geometric renormalizations, and it also extends a result
of Yu to the scale-critical regime.
