Superradiant Bose--Einstein condensates around Kerr black holes

Ultralight bosonic dark matter can accumulate around a rotating black hole, where superradiance amplifies the field until a macroscopic cloud forms. Whether such a cloud behaves as a Bose--Einstein condensate depends on the self-interaction, which earlier work has either retained on static backgrounds or dropped on the Kerr metric. Here we treat the two together. Starting from the Klein--Gordon equation with a quartic potential, we separate the linear problem into spheroidal and radial equations and solve them self-consistently, obtain the superradiant growth rate from the conserved Noether current, project the nonlinear term onto a single mode, and integrate the resulting Gross--Pitaevskii equation with a bordered Newton method at fixed particle number. Rotation modulates the self-interaction geometrically: the effective coupling carries a factor $\Delta(r)$ and therefore switches off at the horizon. Once the field is rescaled, the whole solution family depends on the single dimensionless parameter $\mathcal{N}=\lambda N$. We recover the hydrogenic spectrum of the gravitational atom and the $\alpha^{4\ell+5}$ scaling of the growth rate, and we obtain the exact relation $J_{z}=\hbar mN$, which receives no correction from the self-interaction. We find that the condensate is a torus rather than a spherical shell, with its density vanishing identically on the rotation axis, and that the cloud is modified appreciably only for $\mathcal{N}\gtrsim10^{3}$.