Horizon stability in asymptotically flat spacetimes

It is shown that, provided the ideal point structure of spacetime is stable, generic infinitesimal perturbations cause singularities to appear on the event horizons in both the maximally extended Schwarzschild spacetime and the maximally extended Reissner-Nordstr\"om spacetime. Furthermore, singularities will also appear on the "inner event horizon" of the maximally extended Reissner-Nordstr\"om solution when this spacetime is perturbed generically. The Cauchy horizon appearing in the Reissner-Nordstr\"om solution is shown to be unstable to spherically symmetric infinitesimal perturbations. The barrier that prevents travel through a wormhole from one asymptotically flat region to another is shown in the Schwarzschild case to be stable to spherically symmetric perturbations.