The Scattering Map on Collapsing Charged Spherically Symmetric Spacetimes

TL;DR

This paper extends previous scattering results to charged collapsing spacetimes, demonstrating energy boundedness of wave solutions and analyzing the properties of the scattering map, with implications for Hawking radiation.

Contribution

It generalizes scattering analysis to charged, collapsing spherically symmetric spacetimes, including extremal cases, and studies the boundedness and surjectivity of the scattering map.

Findings

Energy of wave solutions remains uniformly bounded in time.

The scattering map is bounded forward in time but not backward.

No backward scattering map exists for finite energy radiation fields.

Abstract

In this paper we generalise our previous results [1] concerning scattering on the exterior of collapsing dust clouds to the charged case, including in particular the extremal case. We analyse the energy boundedness of solutions $\phi$ to the wave equation on the exterior of collapsing spherically symmetric charged matter clouds. We then proceed to define the scattering map on this spacetime, and look at the implications of our boundedness results on this map. More specifically, we first construct a class of spherically symmetric charged collapsing matter cloud exteriors, and then consider solutions to the wave equation with Dirichlet (reflective) boundary conditions on the surface of these clouds. We then show that the energy of $\phi$ remains uniformly bounded going forwards or backwards in time, and that the scattering map is bounded going forwards but not backwards. Therefore, the scattering map is not surjective onto the space of finite energy on $\mathcal{I}^+\cup\mathcal{H}^+$. Thus there does not exist a backwards scattering map from finite energy radiation fields on $\mathcal{I}^+\cup\mathcal{H}^+$ to finite energy radiation fields on $\mathcal{I}^-$ for these models. These results will be used to give a treatment of Hawking radiation in a companion paper [2].