Solutions to the mode equation for a quantized massless scalar field outside a black hole that forms from the collapse of a null shell: Late-time behaviors and computation of the stress-energy tensor

The late-time behaviors of the modes in the Unruh state for various eternal two-dimensional black holes are discussed. The Unruh state is designed to be a state that mimics the late-time behaviors of quantized fields that Hawking predicted for the black holes that form from collapse. Evidence is provided that the late-time behaviors of some of the modes of the quantum fields and the symmetric two-point function are determined by infrared effects. In four dimensions, a spacetime is considered in which a black hole forms from the collapse of a spherically symmetric null shell. A method is developed to compute the semi-classical stress-energy tensor for a massless minimally-coupled scalar field in the $in$ vacuum state outside the event horizon of the black hole. The computation of the stress-energy tensor involves finding the mode solutions of the scalar field and renormalizing the stress-energy tensor. The former has been done by expanding the modes of the scalar field in terms of a complete set of modes for an eternal Schwarzschild black hole. The latter is done by subtracting the stress-energy tensor for the scalar field in the Unruh state. The computation of the mode solutions of the scalar field in the $in$ vacuum state is discussed. The results are shown for three cases. They are the continuity of the modes for the scalar field across the null shell surface, their high-frequency behaviors for a fixed spacetime point, and their late-time behaviors for a fixed frequency and a fixed spatial point. The preliminary results for the contribution to the stress-energy tensor from the part of the scalar field modes that are ingoing inside the null shell are presented and an approximation for the contribution to the stress-energy tensor from the part of the modes that are outgoing inside the null shell is discussed.