Proof of linear instability of the Reissner-Nordstr\"om Cauchy horizon under scalar perturbations

TL;DR

This paper proves that scalar perturbations in Reissner-Nordström black holes lead to infinite energy at the Cauchy horizon, indicating linear instability and supporting the strong cosmic censorship conjecture.

Contribution

It establishes the generic linear instability of the Reissner-Nordström Cauchy horizon under scalar perturbations, linking to the blue shift effect and Price's law decay.

Findings

Solutions have infinite energy near the Cauchy horizon.

The instability is related to the blue shift effect.

Price's law decay is sharp along the event horizon.

Abstract

It has long been suggested that solutions to linear scalar wave equation $$\Box_g\phi=0$$ on a fixed subextremal Reissner-Nordstr\"om 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 $W^{1,2}_{loc}$. 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-Nordstr\"om 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.