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

Understand

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.