Classical mass inflation versus semiclassical inner horizon inflation

We analyze the geometry of a spherically symmetric black hole with an inner and outer apparent horizon that is perturbed by spherical null shells of matter. On a classical level we observe that the mass inflation instability is triggered, resulting in a growth of curvature and an inward displacement of the inner horizon. We study in detail the inner structure of the mass-inflated region and compare it with previous results obtained for the case in which the perturbing matter content has a continuous distribution. We then perform an approximate calculation of the renormalized stress-energy tensor of a quantum field in the vicinity of the inner horizon, and analyze the semiclassical backreaction on this region of the geometry. We find that the classical tendency for this horizon to move inward due to mass inflation is challenged and potentially overcome by a semiclassical tendency for it to inflate outward.