The Scattering Map on Oppenheimer–Snyder Space-Time

Abstract

AbstractIn this paper, we analyse the boundedness of solutions $$\phi $$ϕ 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 $$\phi $$ϕ remains uniformly bounded going forwards in time and going backwards in time for both the reflective and the permeating cases. It is then shown 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 $$\mathcal {I}^+\cup \mathcal {H}^+$$I+∪H+. Thus, there does not exist a backwards scattering map from finite energy radiation fields on $$\mathcal {I}^+\cup \mathcal {H}^+$$I+∪H+ to finite energy radiation fields on $$\mathcal {I}^-$$I-. We will then contrast this with the situation for scattering in pure Schwarzschild.