Outgoing gravitational shock-wave at the inner horizon: The late-time limit of black hole interiors

We investigate the interiors of 3+1 dimensional asymptotically flat charged and rotating black holes as described by observers who fall into the black holes at late times, long after any perturbations of the exterior region have decayed. In the strict limit of late infall times, the initial experiences of such observers are precisely described by the region of the limiting stationary geometry to the past of its inner horizon. However, we argue that late infall-time observers encounter a null shockwave at the location of the would-be outgoing inner horizon. In particular, for spherically symmetric black hole spacetimes we demonstrate that freely-falling observers experience a metric discontinuity across this shock, that is, a gravitational shock-wave. Furthermore, the magnitude of this shock is at least of order unity. A similar phenomenon of metric discontinuity appears to take place at the inner horizon of a generically-perturbed spinning black hole. We compare the properties of this null shockwave singularity with those of the null weak singularity that forms at the Cauchy horizon.