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Rotating black holes can support quasi-stationary (unstable) bound-state resonances of massive scalar fields in their exterior regions.
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We use natural units in which G = c = ℏ = 1 G=c=\hbar=1
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Note that the field mass parameter μ \mu stands for μ / ℏ \mu/\hbar . Hence, it has the dimensions of ( ( length OPEN ) − 1 )^{-1}
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Here the integer n = 0 , 1 , 2 , … n=0,1,2,... is the resonance parameter of the radial field mode, see Eq. ( 18
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More explicitly, by recording the temporal dependence of the scalar-field amplitude A ( t ) A(t) at some fixed point r = r 0 r=r_{0} , and taking the Fourier transform of this time-dependent field amplitude, one finds Dol2 that the resulting Fourier power spectrum P ( ω ) P(\omega) is characterized by sharp picks at the appropriate resonant frequencies ω R ( μ ) \omega_{\text{R}}(\mu) [see Eq. ( 24
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Here ( t , r , θ , ϕ ) (t,r,\theta,\phi) are the Boyer-Lindquist coordinates Kerr
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Here ω , l \omega,l , and m m are respectively the (conserved) frequency of the field mode, its spheroidal harmonic index, and its azimuthal harmonic index
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