  The motion of gravitational axion-like particles (ALP) around a Kerr black
hole is analyzed, paying attention to resonance and distribution of spectral
radiation. We first discuss the computation of $\sqrt{g}{\tilde R}_{\mu \nu
\rho \rho \sigma}R^{\mu \nu \rho \sigma}$ and its implications with
Pontryagin's theorem and a detailed analysis of Teukolsky's master equation is
done. After carefully analyzing the Teukolsky master equation, we show that
this system exhibits resonance when $\omega \gtrsim \mu$ where $\mu$ is the
mass of the ALP. A skew-normal distribution can approximate the energy
distribution, and we can calculate the mean lifetime of the resonance for black
holes with masses between 100 to 1000 $M_{\odot}$. This range corresponds to a
duration between $10^{-1}$s and $10^{41}$s, the observation range used in LIGO
data.
