A bound on energy extraction (and hairiness) from superradiance

The possibility of mining the rotational energy from black holes has far--reaching implications. Such extraction could occur even for isolated holes, if hypothetical ultralight bosonic particles exist in Nature, leading to a new equilibrium state $-$ hole with synchronised hair whose lifetime exceed age Universe. A natural question is then: an and at maximal efficiency, how large fraction $\epsilon$ that can be extracted Kerr by superradiant growth dominant mode? In other words, hairy resulting become? thermodynamical bound total superradiance $\epsilon\lesssim 0.29$ (as initial mass), long been known, area law. However, numerical simulations exhibiting mode only reached about one third this value. We show development instabilities approximately conservative suggest evolutions), efficiency limited 0.10$, regardless spin field. This agreement maximum obtained vector field predicts result similar scalar field, yet performed.