Trapped Surface Formation for Spherically Symmetric Einstein–Maxwell-Charged Scalar Field System with Double Null Foliation

In this paper, we generalize a method introduced by Christodoulou for studying the Einstein-scalar field to prove trapped surface formation criterion Einstein–Maxwell-charged scalar system under spherical symmetry. If further require initial charge be sufficiently small, obtain an almost-scale-critical result in perturbative regime. “Appendix”, also include proof of Christodoulou’s on using double null coordinates, as well strengthed scale-critical case Minkowskian incoming characteristic data.