This paper shows that a commonly used semiclassical model of gravity can develop a spacelike “thunderbolt” singularity that appears after a black hole’s apparent horizon has receded and stretches out into regions where spacetime curvature was expected to be small. In plain terms, when the authors include quantum effects of matter fields on the classical gravitational field, their equations sometimes produce a sudden, large breakdown far from the black hole. The result suggests the usual semiclassical picture may fail over large distances during black hole evaporation.

The authors study a four‑dimensional semiclassical model that keeps coordinate‑invariance (diffeomorphism invariance) and reproduces the one‑loop trace anomaly, a standard quantum effect that changes the energy‑momentum balance of matter. They work with the Riegert anomaly‑induced action coupled to Einstein gravity. To model collapse they follow a spherically symmetric in‑falling null shell (a simple, sharp pulse of energy) and numerically solve the quantum‑corrected Einstein equations with back‑reaction, so the emitted radiation can affect the spacetime geometry.

Solving those equations is hard because they are fourth‑order, nonlinear partial differential equations. The team uses a characteristic method in double‑null coordinates and integrates the equations numerically with code translated from Mathematica to MATLAB, supplying initial data based on the classical Kruskal black hole solution. Their solutions show a spacelike thunderbolt singularity that forms after the apparent horizon shrinks away and that extends into the exterior region. The authors argue this behavior comes from a nonlinear instability tied to the higher‑derivative (fourth‑order) structure of the semiclassical equations, and they present both numerical evidence and analytic arguments that the effect is generic in models built from anomaly‑induced quantum corrections.