A scattering theory approach to Cauchy horizon instability and applications to mass inflation

Abstract: Motivated by the strong cosmic censorship conjecture, we study the linear scalar wave equation in the interior of subextremal strictly charged Reissner-Nordstr\"om black holes by analyzing a suitably-defined "scattering map" at $0$ frequency. The method can already be demonstrated in the case of spherically symmetric scalar waves on Reissner-Nordstr\"om: we show that assuming suitable ($L^2$-averaged) upper and lower bounds on the event horizon, one can prove ($L^2$-averaged) polynomial lower bound for the solution (1) on any radial null hypersurface transversally intersecting the Cauchy horizon, and (2) along the Cauchy horizon towards timelike infinity. Taken together with known results regarding solutions to the wave equation in the exterior, (1) above in particular provides yet another proof of the linear instability of the Reissner-Nordstr\"{o}m Cauchy horizon. As an application of (2) above, we prove a conditional mass inflation result for a nonlinear system, namely, the Einstein-Maxwell-(real)-scalar field system in spherical symmetry. For this model, it is known that for a generic class of Cauchy data $\mathcal G$, the maximal globally hyperbolic future developments are $C^2$-future-inextendible. We prove that if a (conjectural) improved decay result holds in the exterior region, then for the maximal globally hyperbolic developments arising from initial data in $\mathcal G$, the Hawking mass blows up identically on the Cauchy horizon.