The linear stability of Reissner-Nordstr\"om spacetime for small charge

In this paper, we prove the linear stability to gravitational and electromagnetic perturbations of the Reissner-Nordstr\"om family of charged black holes with small charge. Solutions to the linearized Einstein-Maxwell equations around a Reissner-Nordstr\"om solution arising from regular initial data remain globally bounded on the black hole exterior and in fact decay to a linearized Kerr-Newman metric. We express the perturbations in geodesic outgoing null foliations, also known as Bondi gauge. To obtain decay of the solution, one must add a residual pure gauge solution which is proved to be itself controlled from initial data. Our results rely on decay statements for the Teukolsky system of spin $\pm2$ and spin $\pm1$ satisfied by gauge-invariant null-decomposed curvature components, obtained in earlier works. These decays are then exploited to obtain polynomial decay for all the remaining components of curvature, electromagnetic tensor and Ricci coefficients. In particular, the obtained decay is optimal in the sense that it is the one which is expected to hold in the non-linear stability problem.