Ergosurfaces for Kerr black holes with scalar hair

Abstract

We have recently reported the existence of Kerr black holes with scalar hair in General Relativity minimally coupled to a massive, complex scalar field [C. Herdeiro and E. Radu, Phys. Rev. Lett. 112, 221101 (2014)]. These solutions interpolate between boson stars and Kerr black holes. The latter have a well-known topologically ${S}^{2}$ ergosurface (ergosphere) whereas the former develop a ${S}^{1}\ifmmode\times\else\texttimes\fi{}{S}^{1}$ ergosurface (ergotorus) in a region of parameter space. We show that hairy black holes always have an ergoregion, and that this region is delimited by either an ergosphere or an ergo-Saturn---i.e. a ${S}^{2}\ensuremath{\bigoplus}({S}^{1}\ifmmode\times\else\texttimes\fi{}{S}^{1})$ ergosurface. In the phase space of solutions, the ergotorus can either appear disconnected from the ergosphere or pinch off from it. We provide a heuristic argument, based on a measure of the size of the ergoregion, that superradiant instabilities---which are likely to be present---are weaker for hairy black holes than for Kerr black holes with the same global charges. We observe that Saturn-like, and even more remarkable, ergosurfaces should also arise for other rotating ``hairy'' black holes.