Spherical electro-vacuum black holes with resonant, scalar Q-hair

The asymptotically flat, spherical, electro-vacuum black holes (BHs) are shown to support static, spherical configurations of a gauged, self-interacting, scalar field, minimally coupled to the geometry. Considering a Q -ball type potential for the scalar field, we dub these configurations Q - clouds , in the test field approximation. The clouds exist under a resonance condition , at the threshold of (charged) superradiance. This is similar to the stationary clouds supported by Kerr BHs, which exist for a synchronisation condition , at the threshold of (rotational) superradiance. In contrast with the rotating case, however, Q -clouds require the scalar field to be massive and self-interacting; no similar clouds exist for massive but free scalar fields. First, considering a decoupling limit, we construct Q -clouds around Schwarzschild and Reissner–Nordström BHs, showing there is always a mass gap. Then, we make the Q -clouds backreact, and construct fully non-linear solutions of the Einstein–Maxwell-gauged scalar system describing spherical, charged BHs with resonant, scalar Q - hair . Amongst other properties, we observe there is non-uniqueness of charged BHs in this model and the Q -hairy BHs can be entropically preferred over Reissner–Nordström, for the same charge to mass ratio; some Q -hairy BH solutions can be overcharged. We also discuss how some well known no-hair theorems in the literature, applying to electro-vacuum plus minimally coupled scalar fields, are circumvented by this new type of BHs.