Vacuum Energy at Apparent Horizon in Conventional Model of Black Holes

For a black hole of Schwarzschild radius $a$, we argue that the back-reaction of the vacuum energy-momentum tensor is in general important at the apparent horizon when the time scale of a process is larger than order $a$. In particular, in a double-shell model, we show that the ignorance of the back-reaction leads to a divergence in the outgoing vacuum energy flux. The main result of this paper is that, once the back-reaction is included, the vacuum energy density for observers on top of the trapping horizon in vacuum is given by ${\cal E} \simeq -1/2\ell_p^2 a^2$, which is of the same order of magnitude as the classical matter but opposite in sign. Remarkably, this formula is independent of both the details about the collapsing matter and the vacuum energy-momentum tensor.