Abstract
In this paper, we prove the linear stability to gravitational and electromagnetic perturbations of the Reissner–Nordström family of charged black holes with small charge. Solutions to the linearized Einstein-Maxwell equations around a Reissner-Nordström 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 \(\pm \,2\) and spin \(\pm \,1\) 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.
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Acknowledgements
The author would like to thank Sergiu Klainerman and Mu-Tao Wang for their guidance and support. The author is also grateful to Jérémie Szeftel, Pei-Ken Hung and Federico Pasqualotto for helpful discussions. The author is grateful to the anonymous referee for several helpful suggestions.
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Giorgi, E. The Linear Stability of Reissner–Nordström Spacetime for Small Charge. Ann. PDE 6, 8 (2020). https://doi.org/10.1007/s40818-020-00082-y
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DOI: https://doi.org/10.1007/s40818-020-00082-y