Quasi-bound states of massive scalar fields in the Kerr black-hole spacetime: Beyond the hydrogenic approximation

Rotating black holes can support quasi-stationary (unstable) bound-state resonances of massive scalar fields in their exterior regions. These spatially regular scalar configurations are characterized by instability timescales which are much longer than the timescale M set by the geometric size (mass) of the central black hole. It is well-known that, in the small-mass limit alpha = M mu << 1(here mu is the mass of the scalar field), these quasi-stationary scalar resonances are characterized by the familiar hydrogenic oscillation spectrum: omega(R)/mu = 1 - alpha(2)/2 (n) over bar (2)(0), where the integer (n) over bar (0)(l, n; alpha -> 0) = l + n + 1 is the principal quantum number of the bound-state resonance (here the integers l = 1, 2, 3, ... and n = 0, 1, 2, ... are the spheroidal harmonic index and the resonance parameter of the field mode, respectively). As it depends only on the principal resonance parameter (n) over bar (0), this small-mass (alpha << 1) hydrogenic spectrum is obviously degenerate. In this paper we go beyond the small-mass approximation and analyze the quasi-stationary bound-state resonances of massive scalar fields in rapidly-spinning Kerr black-hole spacetimes in the regime alpha = O(1). In particular, we derive the non-hydrogenic (and, in general, non-degenerate) resonance oscillation spectrum omega(R)/mu = root 1 - (alpha/(n) over bar)(2), where (n) over bar (l, n; alpha) = root(l + 1/2)(2) - 2m alpha + 2 alpha(2) + 1/2 + n is the generalized principal quantum number of the quasi-stationary resonances. This analytically derived formula for the characteristic oscillation frequencies of the composed black-hole-massive-scalar-field system is shown to agree with direct numerical computations of the quasi-stationary bound-state resonances. (C) 2015 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license