PROOF OF LINEAR INSTABILITY OF THE REISSNER-NORDSTROM CAUCHY HORIZON UNDER SCALAR PERTURBATIONS

It has long been suggested that solutions to the linear scalar wave equation square(g)phi = 0 on a fixed subextremal Reissner-Nordstrom spacetime with nonvanishing 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 Wolf. 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-Nordstrom 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.