The linear stability of Reissner-Nordström spacetime for small charge

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.

arXiv comment121 pages, 2 figures, version accepted for publication

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• Decay of the Local Energy for the Charged Klein-Gordon Equation in the Exterior De Sitter-Reissner-Nordström Spacetime
• Static conformal elastic solution of Einstein's field equations
• Numerical investigation of the late-time tails of the solutions of the Fackerell-Ipser equation
• Physical-space estimates for axisymmetric waves on extremal Kerr spacetime