### 3.5 Electric charge

Given the scaling power law for the black hole mass in critical collapse, one would like to know what
happens if one takes a generic 1-parameter family of initial data with both electric charge and
angular momentum (for suitable matter), and fine-tunes the parameter p to the black hole
threshold. A simple model for charged matter is a complex scalar field coupled to electromagnetism
with the substitution , or scalar electrodynamics. (Note that in geometric
units, black hole charge Q has dimension of length, but the charge parameter e has dimension
1/length.)
Gundlach and Martín-García [106] 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

was predicted. The predicted value of the critical exponent (in scalar electrodynamics) was
subsequently verified in collapse simulations by Hod and Piran [126] and later again by Petryk [176]. (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 [176].
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

(with the equality holding if the critical solution is charged), so that black hole charge can always be treated
perturbatively at the black hole threshold. This has not yet been verified in any other matter
model.