In this paper we use Morawetz and geometric energy estimates—the so-called vector field method—to prove decay results for the Maxwell field in the static exterior region of the Reissner–Nordström–de Sitter black hole. We prove two types of decay: the first is a uniform decay of the energy of the Maxwell field on achronal hypersurfaces as the hypersurfaces approach timelike infinities. The second decay result is a pointwise decay in time with a rate of $$t^{-1}$$ t - 1 which follows from local energy decay by Sobolev estimates. Both results are consequences of bounds on the conformal energy defined by the Morawetz conformal vector field. These bounds are obtained through wave analysis on the middle spin component of the field. The results hold for a more general class of spherically symmetric spacetimes with the same arguments used in this paper.

中文翻译：

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Reissner-Nordström-de Sitter 黑洞上麦克斯韦场的衰减

在本文中，我们使用 Morawetz 和几何能量估计（即所谓的矢量场方法）来证明 Reissner-Nordström-de Sitter 黑洞静态外部区域中麦克斯韦场的衰减结果。我们证明了两种类型的衰减：第一种是随着超曲面接近类时无穷大，麦克斯韦场在非时态超曲面上的能量均匀衰减。第二个衰减结果是时间上的逐点衰减，其速率为 $$t^{-1}$$ t - 1，其源自 Sobolev 估计的局部能量衰减。这两个结果都是由 Morawetz 共形矢量场定义的共形能量边界的结果。这些边界是通过对场的中间自旋分量进行波分析获得的。结果适用于更一般的球对称时空类，其参数与本文中使用的参数相同。