Kerr black holes with synchronised scalar hair and higher azimuthal harmonic index

Kerr black holes with synchronised scalar hair and azimuthal harmonic index m > 1 are constructed and studied. The corresponding domain of existence has a broader frequency range than the fundamental m = 1 family; moreover, larger ADM masses, M and angular momenta J are allowed. Amongst other salient features, non-uniqueness of solutions for fixed global quantities is observed: solutions with the same M and J co-exist, for consecutive values of m, and the ones with larger m are always entropically favoured. Our analysis demonstrates, moreover, the qualitative universality of various features observed for m = 1 solutions, such as the shape of the domain of existence, the typology of ergo-regions, and the horizon geometry, which is studied through its isometric embedding in Euclidean 3-space. (C) 2019 The Authors. Published by Elsevier B.V.