Self-consistent evaporation of a 4D black hole including back-reaction
In plain words
Most evaporation calculations assume the black hole shrinks slowly while keeping its shape. A full four-dimensional calculation in which the radiation reshapes spacetime as it goes has not been done.
Precise statement
Solve the semiclassical Einstein equations with the renormalized $\langle T_{ab}\rangle$ of $N_{f}$ free massless fields (all angular momenta, not only s-waves) for a black hole formed by collapse, from formation to $M \sim m_{P}$, with $N_{f}$ large to suppress fluctuations. Determine the evolution of the apparent and event horizons, whether an inner apparent horizon forms and where the trapped region closes, and the deviation of $\mathrm{d}M/\mathrm{d}t$ from the adiabatic rate. An answer is a numerical or analytic spacetime with controlled errors.
What would settle it
A convergent numerical solution of the 4D semiclassical equations through most of the evaporation with the full renormalized stress tensor.
Status in the literature
Unverified note
Fully back-reacted solutions exist in 2D dilaton models such as CGHS (Ashtekar, Pretorius, Ramazanoglu 2011); the 4D problem with the complete stress tensor is open as of 2026.