The Scattering Map on Oppenheimer--Snyder Spacetime

TL;DR

This paper investigates the boundedness of wave solutions and the properties of the scattering map in the Oppenheimer--Snyder gravitational collapse spacetime, revealing that the scattering map is bounded forward but not backward, with implications for wave propagation.

Contribution

It introduces the analysis of the scattering map in Oppenheimer--Snyder spacetime and demonstrates its boundedness properties, contrasting with Schwarzschild scattering.

Findings

Energy of wave solutions remains uniformly bounded over 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 analyse the boundedness of solutions $\phi$ of the wave equation in the Oppenheimer--Snyder model of gravitational collapse in both the case of a reflective dust cloud and a permeating dust cloud. We then proceed to define the scattering map on this space-time, and look at the implications of our boundedness results on this scattering map. Specifically, it is shown that the energy of $\phi$ remains uniformly bounded going forward in time and going backwards in time for both the reflective and the permeating cases, and it is then shown that the scattering map is bounded going forwards, but not backwards, and is therefore 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 $\mathcal{I}^-$. We will then contrast this with the situation for scattering in pure Schwarzschild.