We derive the precise late-time asymptotics for solutions to the wave equation on extremal Reissner-Nordstr\"om black holes and explicitly express the leading-order coefficients in terms of the initial data. Our method is based on purely physical space techniques. We derive novel weighted energy hierarchies and develop a singular time inversion theory, which allow us to uncover the subtle contribution of both the near-horizon and near-infinity regions to the precise asymptotics. We introduce a new horizon charge and provide applications pertaining to the interior dynamics of extremal black holes. Our work confirms, and in some cases extends, the numerical and heuristic analysis of Lucietti-Murata-Reall-Tanahashi, Ori-Sela and Blaksley-Burko.

中文翻译：

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极值 Reissner-Nordström 背景下波动方程的后期渐近性

我们推导出了极值 Reissner-Nordstr\"om 黑洞波动方程解的精确后期渐近线，并根据初始数据明确表达了前级系数。我们的方法基于纯物理空间技术。我们推导出新的加权能量层次并发展奇异时间反演理论，这使我们能够揭示近视距和近无穷远区域对精确渐近的微妙贡献。我们引入了新的视界电荷并提供了与内部有关的应用极值黑洞的动力学。我们的工作证实并在某些情况下扩展了 Lucietti-Murata-Reall-Tanahashi、Ori-Sela 和 Blaksley-Burko 的数值和启发式分析。