Proof of linear instability of the Reissner-Nordström Cauchy horizon under scalar perturbations

Abstract

It has long been suggested that solutions to linear scalar wave equation $${\square }_g\varphi =0$$ on a fixed subextremal Reissner-Nordström spacetime with non-vanishing charge are generically singular at the Cauchy horizon. We prove that generic smooth and compactly supported initial data on a Cauchy hypersurface indeed give rise to solutions with infinite nondegenerate energy near the Cauchy horizon in the interior of the black hole. In particular, the solution generically does not belong to $W_{loc}^{1,2}$. This instability is related to the celebrated blue shift effect in the interior of the black hole. The problem is motivated by the strong cosmic censorship conjecture and it is expected that for the full nonlinear Einstein-Maxwell system, this instability leads to a singular Cauchy horizon for generic small perturbations of Reissner-Nordström spacetime. Moreover, in addition to the instability result, we also show as a consequence of the proof that Price's law decay is generically sharp along the event horizon.

References in corpus (5)

• Hidden symmetries and decay for the Vlasov equation on the Kerr spacetime
• A scattering theory for the wave equation on Kerr black hole exteriors
• The wave equation on Schwarzschild-de Sitter spacetimes
• Price's Law on Nonstationary Spacetimes
• Decay for solutions of the wave equation on Kerr exterior spacetimes I-II: The cases |a| << M or axisymmetry

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