LAUSR

Assuming that the vacuum energy-momentum tensor is not exceptionally large, we give a general proof for 4D evaporating black holes with spherical symmetry that the surface of the collapsing matter can never be farther inside the timelike trapping horizon than a proper distance $\sim \mathcal{O}(n^{3/2}\ell_p)$ when the black hole is evaporated to $1/n$ of its initial mass, as long as $n \ll a^{2/3}/\ell_p^{2/3}$ (where $a$ is the Schwarzschild radius and $\ell_p$ is the Planck length). For example, the distance between the matter and the apparent horizon must be Planckian at the Page time.