AI 中文总结
本文结合克尔度规与自相互作用，求解超轻玻色子暗物质在克尔黑洞周围的超辐射凝聚问题，得到环形凝聚解及相关标度关系，揭示旋转对自相互作用的调制效应。
AI 中文摘要
超轻玻色子暗物质会在旋转黑洞周围聚集，超辐射效应会放大该场直至形成宏观云团。这类云团是否表现为玻色-爱因斯坦凝聚体取决于自相互作用，此前研究要么在静态背景下保留自相互作用，要么在克尔度规下忽略自相互作用。本文将二者结合研究：从带四次势的克莱因-戈登方程出发，将线性问题分离为椭球方程和径向方程并自洽求解，从守恒的诺特定流得到超辐射增长率，将非线性项投影到单一模式上，在固定粒子数下用带边界的牛顿法求解得到的格罗斯-皮塔耶夫斯基方程。旋转会从几何上调制自相互作用：有效耦合带有因子Δ(r)，因此在视界处消失。场经重新标度后，整个解族仅依赖无量纲参数𝒩=λN。我们得到了引力原子的类氢光谱和增长率的α^(4ℓ+5)标度关系，还得到精确关系J_z=ħmN，该关系不受自相互作用修正。研究发现，凝聚体为环形而非球壳，其密度在旋转轴上完全消失，且仅当𝒩≳10³时，云团才会发生显著修正。
英文摘要
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 $\mathcal{N}=λN$. We recover the hydrogenic spectrum of the gravitational atom and the $α^{4\ell+5}$ 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 $\mathcal{N}\gtrsim10^{3}$.
Comments38 pages, 11 figures
