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 ∇ → ∇ + ieA a a a, 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 [106Jump To The Next Citation Point] 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

Q ∼ (p − p )δ (33 ) ∗
was predicted. The predicted value δ ≃ 0.88 of the critical exponent (in scalar electrodynamics) was subsequently verified in collapse simulations by Hod and Piran [126Jump To The Next Citation Point] and later again by Petryk [176Jump To The Next Citation Point]. (The mass scales with γ ≃ 0.37 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 |Q| ≲ M can be formed [176Jump To The Next Citation Point].

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

δ ≥ 2γ (34 )
(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.
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