Massive bosons in the vicinity of Kerr–Newman black holes can form pure bound states when their phase angular velocity fulfills the synchronization condition, i.e. at the threshold of superradiance. The presence of these stationary clouds at the linear level is intimately linked to the existence of Kerr black holes with synchronized hair at the nonlinear level. These configurations are very similar to the atomic orbitals of the electron in a hydrogen atom. They can be labeled by four quantum numbers: [Formula: see text], the number of nodes in the radial direction; [Formula: see text], the orbital angular momentum; [Formula: see text], the total angular momentum; and [Formula: see text], the azimuthal total angular momentum. These synchronized configurations are solely allowed for particular values of the black holes mass, angular momentum and electric charge. Such quantization results in an existence surface in the three-dimensional parameter space of Kerr–Newman black holes. The phenomenology of stationary scalar clouds has been widely addressed over the last years. However, there is a gap in the literature concerning their vector cousins. Following the separability of the Proca equation in Kerr(–Newman) spacetime, this paper explores and compares scalar and vector stationary clouds around Kerr and Kerr–Newman black holes, extending previous research.

中文翻译：

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克尔-纽曼黑洞周围的静止标量云和矢量云

Kerr-Newman 黑洞附近的大质量玻色子在其相位角速度满足同步条件时，即处于超辐射阈值时，可以形成纯束缚态。这些静止云在线性水平上的存在与在非线性水平上具有同步头发的克尔黑洞的存在密切相关。这些构型与氢原子中电子的原子轨道非常相似。它们可以用四个量子数来标记：【公式：见正文】，径向上的节点数；【公式：见正文】，轨道角动量；【公式：见正文】，总角动量；[公式：见正文]，方位角总角动量。这些同步配置仅允许用于特定的黑洞质量值，角动量和电荷。这种量化导致在克尔-纽曼黑洞的三维参数空间中存在一个存在面。静止标量云的现象学在过去几年中得到了广泛的解决。然而，关于它们的向量表亲的文献存在空白。根据Proca方程在Kerr(-Newman)时空的可分性，本文探索和比较了Kerr和Kerr-Newman黑洞周围的标量和矢量静止云，扩展了之前的研究。