2026/08/03 by Sen Guo, Lin Wen, Xiao-Xiong Zeng
Physics and Astronomy · #gr-qc
38 pages, 11 figures
arxiv created 2026/08/03 · arxiv updated 2026/08/04
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 Δ(r) and therefore switches off at the horizon. Once the field is rescaled, the whole solution family depends on the single dimensionless parameter N=λN. We recover the hydrogenic spectrum of the gravitational atom and the α4ℓ+5 scaling of the growth rate, and we obtain the exact relation Jz=ℏ 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 N\gtrsim103.