The Scattering Map on Oppenheimer--Snyder Spacetime

Abstract

In this paper we analyse the boundedness of solutions $\varphi$ 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 $\varphi$ 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.

References in corpus (3)

• The black hole stability problem for linear scalar perturbations
• A new physical-space approach to decay for the wave equation with applications to black hole spacetimes
• Time-Translation Invariance of Scattering Maps and Blue-Shift Instabilities on Kerr Black Hole Spacetimes

Cited by in corpus (6)

• A Scattering Theory for Linearised Gravity on the Exterior of the Schwarzschild Black Hole I: The Teukolsky Equations
• Strong Cosmic Censorship in the presence of matter: the decisive effect of horizon oscillations on the black hole interior geometry
• Quantum Field Theory in Curved Spacetime (2nd Edition)
• Quantum oscillations in the black hole horizon
• Conformal Scattering of Maxwell Potentials
• Conformal scattering theories for tensorial wave equations on Schwarzschild spacetime