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

Understand

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.