  A method is presented which allows for the numerical computation of the
stress-energy tensor for a quantized massless minimally coupled scalar field in
the region outside the event horizon of a 4D Schwarzschild black hole that
forms from the collapse of a null shell. This method involves taking the
difference between the stress-energy tensor for the in state in the collapsing
null shell spacetime and that for the Unruh state in Schwarzschild spacetime.
The construction of the modes for the in vacuum state and the Unruh state is
discussed. Applying the method, the renormalized stress-energy tensor in the 2D
case has been computed numerically and shown to be in agreement with the known
analytic solution. In 4D, the presence of an effective potential in the mode
equation causes scattering effects that make the construction of the in modes
more complicated. The numerical computation of the in modes in this case is
given.
