Amplification of gravitational wave by a Kerr black hole

Binary black hole may form near a supermassive black hole. The background
black hole (BH) will affect the gravitational wave (GW) generated by the binary
black hole. It is well known that the Penrose process may provide extra energy
due to the ergosphere. In the present paper we investigate the energy
amplification of the gravitational wave by a Kerr black hole background. In
particular and different from the earlier studies, we compare the energies of
the waves in the cases with and without a nearby Kerr BH. We find that only
when the binary black hole is moving relative to the Kerr background can the GW
energy be amplified. Otherwise, the energy will be suppressed by the background
Kerr black hole. This finding is consistent with the inequality found by Wald
for Penrose process. Taking into account realistic astrophysical scenarios, we
find that the Kerr black hole background can amplify the GW energy by at most 5
times.