We exhibit an explicit foliation of Schwarzschild space–time by spacelike hypersurfaces which extend from Schwarzschild r=0 to r=∞. This allows us to compute the values of a massless scalar field for all space–time points which lie in the future of the surface on which we initially quantize the theory. This is to be contrasted with approaches which start at past null infinity and propagate to future null infinity. One of its virtues is that this method allows us to discuss both asymptotic Hawking radiation and what is happening at finite distances from the black hole. In order to explain the techniques we use, we begin by discussing variants of the flat-space moving mirror9 problem. Then we discuss the canonical quantization of the massless scalar field theory and the geometric optics approximation which we use to solve the Heisenberg equations of motion in the black hole background. Using the example of an infalling mirror, an analogue of the moving mirror problem, we show that, although our spacelike slices extend to r=0, we can avoid discussing an initial state which extends through the horizon. Furthermore, we show that in the same way we avoid having to deal with the singularity at r=0 when we first quantize the system. This discussion naturally leads to a suggestion of how to handle the question of what is happening when the mirror hits the singularity. In the last section of the paper we discuss a discretization of the computation which behaves in the manner we suggest and yet exhibits Hawking radiation.5 This formulation of the problem allows us to discuss all the issues in an explicitly unitary setting. The resulting picture raises some interesting questions about the information paradox.
Self-force of a scalar charge in the space-time of extreme charged anti-dilatonic wormhole
