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

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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 α≡ Mμ≪1 (here μ is the mass of the scalar field), these quasi-stationary scalar resonances are characterized by the familiar hydrogenic oscillation spectrum: ωᵣ/μ=1-α²/2 n²₀, where the integer n₀(l,n;α→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 n₀, this small-mass (α≪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 α=O(1). In particular, we derive the non-hydrogenic (and, in general, non-degenerate) resonance oscillation spectrum ωᵣ/μ=√1-(α/ n)², where n(l,n;α)=√(l+1/2)²-2mα+2α²+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.