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

A scale-critical trapped surface formation criterion for the Einstein-Maxwell system