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 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 \(\mathcal {I}^+\cup \mathcal {H}^+\). 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.

中文翻译：

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Oppenheimer–Snyder时空上的散射图

在本文中，我们分析了在反射性粉尘云和渗透性粉尘云的情况下，Oppenheimer-Snyder重力塌陷模型中波动方程的解\（\ phi \）的有界性。然后，我们继续在此时空上定义散射图，并查看我们的有限度结果在此散射图上的含义。具体地，表明对于反射和渗透情况，\（\ phi \）的能量在时间上向前和向后都保持均匀有界。然后表明，散射图是有界的，但向前没有限制。因此，散射图不排斥在\（\ mathcal {I} ^ + \ cup \ mathcal {H} ^ + \）上的有限能量空间上。因此，不存在从\（\ mathcal {I} ^ + \ cup \ mathcal {H} ^ + \）上的有限能量辐射场到\（\ mathcal {I} ^ -\）。然后，我们将其与纯Schwarzschild中的散射情况进行对比。