We prove boundedness and decay statements for solutions to the spin \(\pm \,2\) generalized Teukolsky system on a Reissner–Nordström background with very small charge. The first equation of the system is the generalization of the Teukolsky equation in Schwarzschild for the extreme component of the curvature \(\alpha \). The second equation, coupled with the first one, is a new equation for a new gauge-invariant quantity involving the electromagnetic curvature components. The proof is based on the use of derived quantities, introduced in previous works on linear stability of Schwarzschild (Dafermos et al. in Acta Math 222:1–214, 2019). These derived quantities are shown to verify a generalized coupled Regge–Wheeler system. These equations are the ones verified by the extreme null curvature and electromagnetic components of a gravitational and electromagnetic perturbation of the Reissner–Nordström spacetime. Consequently, as in the Schwarzschild case, these bounds provide the first step in proving the full linear stability of the Reissner–Nordström metric for small charge to coupled gravitational and electromagnetic perturbations.

中文翻译：

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Teukolsky自旋系统$$ \ pm \，2 $$±2在Reissner–Nordström时空上的有界和衰减：案例$$ | Q | \ ll M $$ | 问| ≪ M

我们证明了在Reissner–Nordström背景下具有很小电荷的自旋\（\ pm \，2 \）广义Teukolsky系统解的有界和衰减陈述。该系统的第一个方程是Schwarzschild中Teukolsky方程的推广，用于曲率\（\ alpha \）的极值部分。第二个方程式与第一个方程式一起，是一个新的方程式，用于包含电磁曲率分量的新的量规不变量。该证明是基于对先前的Schwarzschild线性稳定性的工作中引入的导出数量的使用（Dafermos等人，Acta Math 222：1-214，2019）。这些导出的量显示出可以验证通用的Regge-Wheeler耦合系统。这些方程式由Reissner–Nordström时空的引力和电磁扰动的极零曲率和电磁分量验证。因此，与Schwarzschild案例一样，这些界限为证明Reissner-Nordström度量的完整线性稳定性（为小电荷与重力和电磁扰动耦合）提供了第一步。