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Instability of gravitational and electromagnetic perturbations of extremal Reissner-Nordström spacetime. (English) Zbl 1540.35393

Summary: We study the linear stability problem to gravitational and electromagnetic perturbations of the extremal, \(|Q|=M\), Reissner-Nordström spacetime, as a solution to the Einstein-Maxwell equations. Our work uses and extends the framework [Classical Quantum Gravity 36, No. 20, Article ID 205001, 48 p. (2019; Zbl 1479.83139); Ann. Henri Poincaré 21, No. 8, 2485–2580 (2020; Zbl 1447.83006)] of E. Giorgi, and contrary to the subextremal case we prove that instability results hold for a set of gauge invariant quantities along the event horizon \({\mathcal{H}}^+\). In particular, for associated quantities shown to satisfy generalized Regge-Wheeler equations we prove decay, non-decay, and polynomial blow-up estimates asymptotically along \({\mathcal{H}}^+\), the exact behavior depending on the number of translation invariant derivatives that we take. As a consequence, we show that for generic initial data, solutions to the generalized Teukolsky system of positive and negative spin satisfy both stability and instability results. It is worth mentioning that the negative spin solutions are significantly more unstable, with the extreme curvature component \({\underline{\alpha}}\) not decaying asymptotically along the event horizon \({\mathcal{H}}^+\), a result previously unknown in the literature.

MSC:

35Q75 PDEs in connection with relativity and gravitational theory
83C22 Einstein-Maxwell equations
83C25 Approximation procedures, weak fields in general relativity and gravitational theory
83C57 Black holes
35B44 Blow-up in context of PDEs
35B35 Stability in context of PDEs
35B20 Perturbations in context of PDEs

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