  To see the back reaction of the Hawking radiation in a dynamical spacetime of
the spherical gravitational collapse, we explicitly calculate the weak value of
the energy-momentum tensor of the massless scalar field. The background
geometry of a collapsing dust sphere is specified by using the
Painleve-Gullstrand coordinates, in which the time coordinate coincides with
the proper time of a free-falling observer and the metric tensor is regular at
the event horizon. The result is that in the remote future the weak value
diverges at the event horizon. We argue that since the semi-classical
approximation of the Einstein equation in the sense of the weak value breaks
down there, the future geometry of the spacetime cannot be the Schwarzschild
geometry.
