Gundlach and Martín-García  have studied scalar massless electrodynamics in spherical symmetry perturbatively. Clearly, the real scalar field critical solution of Choptuik is a solution of this system too. In fact, it remains a critical solution within massless scalar electrodynamics in the sense that it still has only one growing perturbation mode within the enlarged solution space. Some of its perturbations carry electric charge, but as they are all decaying, electric charge is a subdominant effect. The charge of the black hole in the critical limit is dominated by the most slowly decaying of the charged modes. From this analysis, a universal power-law scaling of the black hole charge and later again by Petryk . (The mass scales with as for the uncharged scalar field.) No other type of criticality can be found in the phase space of this system as dispersion and black holes are the only possible end states, though black holes with can be formed .
General considerations similar to those in Section 2.6 led Gundlach and Martín-García a to the general prediction that the two critical exponents are always related, for any matter model, by the inequality
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