Local Group Velocity Distribution inside Superradiant Condensates

Superradiance enables scalar fields to extract energy and angular momentum from a rotating black hole (BH), leading to the formation of a BH-condensate system. Previous studies mainly focus on the phase velocity, which propagates in the azimuthal direction. In this work, we show that the superradiant scalar condensate presents a nontrivial group velocity distribution. In the region sufficiently far from the BH, the condensate exhibits a radial velocity magnitude that approaches $(r_g\mu/2) \sin (2\omega t-2 \varphi)$, while the polar and azimuthal velocity magnitudes asymptotically decline as $\propto 1/r$.