克尔黑洞周围的定态狄拉克凝聚体
Stationary Dirac condensates around Kerr black holes
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中文总结 AI 辅助
本文建立框架求解克尔背景下狄拉克场定态束缚态，发现旋转费米子云为扁椭球几何，且因同步轨迹处无奇异分支，黑洞无法支撑定态狄拉克毛发，揭示了黑洞支撑宏观场的新机制。
中文摘要 AI 辅助
极轻玻色子场可在旋转黑洞周围形成宏观云，而类似的定态费米子凝聚体的存在则受到其固有自旋的严格限制。本文建立了完整的几何与运动学框架，以求解克尔（Kerr）和克尔-纽曼（Kerr-Newman）背景下有质量狄拉克场的定态束缚态。通过将克尔-狄拉克系统映射为全局积分的源径向问题，我们严格分离出视界因果性所决定的边界约束。角向部分揭示了一个基本拓扑差异：由于方位角量子数严格为半整数，正则边界分支阻止了旋转轴上场密度的消失。因此，旋转费米子云天生形成全局填充的扁椭球几何，与标量凝聚体特有的空心环面结构形成鲜明对比。至关重要的是，我们的径向指标分析揭示了同步狄拉克毛发缺失的确切数学起源：恰在运动学同步轨迹处，狄拉克算子的弗罗贝尼乌斯（Frobenius）矩阵非亏损，且完全不含对数发散。没有这些需通过边界正则性选择性切除的奇异分支，存在性的物理负担完全落在因果通量垒上，后者严格使零源振幅变得平凡。这一同步否决表明，黑洞支撑宏观定态场的能力不仅由超辐射运动学决定，还由局域视界因果性与量子自旋统计之间的深刻相互作用决定。
英文摘要
Ultralight bosonic fields can form macroscopic clouds around rotating black holes, whereas the existence of analogous stationary fermionic condensates is strictly constrained by their intrinsic spin. Here we establish a complete geometric and kinematic framework to resolve the stationary bound states of massive Dirac fields on Kerr and Kerr-Newman backgrounds. By mapping the Kerr-Dirac system to a globally integrated sourced radial problem, we strictly isolate the boundary constraints dictated by horizon causality. The angular sector reveals a fundamental topological distinction: because the azimuthal quantum number is strictly half-integer, the regular boundary branches prevent the local field density from vanishing on the rotation axis. Consequently, rotating fermionic clouds inherently form globally filled, oblate geometries, in stark contrast to the hollow toroidal structures characteristic of scalar condensates. Crucially, our radial indicial analysis unveils the exact mathematical origin of the absence of synchronized Dirac hair. Precisely at the kinematic synchronization locus, the Frobenius matrix of the Dirac operator is non-defective and entirely devoid of logarithmic divergences. Without these singular branches to be selectively excised by boundary regularity, the physical burden of existence falls entirely onto the causal flux barrier, which strictly trivializes the zero-source amplitude. This synchronization veto demonstrates that a black hole's capacity to support macroscopic stationary fields is governed not merely by superradiant kinematics, but by the profound interplay between local horizon causality and quantum spin statistics.
