Semiclassical regularity of compact trapped regions: From dynamical horizons to inner extremality

In eternal black-hole spacetimes, inner horizons are Cauchy horizons and are generically unstable. For non-extremal inner horizons, this includes both the classical mass-inflation instability and a semiclassical instability associated with divergences in the renormalized stress-energy tensor (RSET). Inner-extremal geometries, for which the inner-horizon surface gravity vanishes, evade classical mass inflation, but in stationary settings still suffer from singular behavior of the RSET. In this work, we show that the dynamical case is qualitatively different. Considering spacetimes describing the formation and evaporation of a compact trapped region in finite time, and working in the $s$-wave Polyakov approximation, we compute the expectation value of the stress-energy tensor in the in-vacuum state. Given that in this case the inner horizon is not a Cauchy horizon, the RSET remains finite everywhere. For generic non-extremal inner horizons, however, the RSET grows exponentially in time at the inner horizon, with a divergence emerging only in the asymptotic limit of an ever-lasting trapped region. For inner-extremal geometries this exponential growth is replaced by a considerably milder power-law growth. Such spacetimes may therefore be considered natural candidates for classically and semiclassically meta-stable black-hole interiors.