极端旋转黑洞事件视界上波的微局部分析
Microlocal Analysis of Waves Across the Event Horizon of an Extremal Rotating Black Hole
AI总结：
针对极端旋转黑洞事件视界微局部理论不完善的问题，证明极端克尔时空视界零余向量构成对合双特征流形并扩展拟基本解构造，为该区域渐近振荡解提供数学基础。
摘要：
静态黑洞包含连光都无法逃逸的时空区域。在质心系中，这些区域被事件视界分隔。与类点粒子不同，场可以在视界上非因果地传播和相互作用。然而，这方面的微局部理论尚不完善。例如，实主型算子理论不适用于视界。本文针对极端旋转黑洞解决这一问题：证明极端克尔时空视界上的零余向量构成对合双特征流形，然后将拟基本解的构造扩展到事件视界之外。这为该区域的渐近振荡解提供了数学基础，此类近似在量子力学中至关重要。与实主型情况不同，视界附近的解存在两个奇异性传播通道，确实存在沿双特征簇的额外传播。
英文摘要：
Static black holes contain regions of spacetime which not even light can escape from. In the centre of mass frame, these blocks are separated from each other by event horizons. Unlike pointlike particles, fields can spread and interact non-causally across the horizons. The microlocal theory of this is somewhat incomplete, however. For instance, the theory of real principal type operators does not apply on the horizon. In this article, we address this issue for the extremal rotating black hole. Namely, we show that null covectors on the horizon of the extremal Kerr spacetime form an involutive double characteristic manifold and then extend the construction of parametrix across the event horizon. This provides a mathematical basis for the asymptotic oscillatory solutions in the region. Such approximations are central in quantum mechanics. In contrast to the real principal case, solutions near the horizon admit two channels for propagation of singularities. Indeed, there is additional propagation along the double characteristic variety.
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