Angelopoulos Yannis, Aretakis Stefanos, Gajic Dejan
The Division of Physics, Mathematics and Astronomy, Caltech, 1200 E California Blvd, Pasadena, CA 91125 USA.
Department of Mathematics, University of Toronto, 40 St George Street, Toronto, ON Canada.
Commun Math Phys. 2020;380(1):323-408. doi: 10.1007/s00220-020-03857-3. Epub 2020 Sep 23.
It is known that sub-extremal black hole backgrounds do not admit a (bijective) non-degenerate scattering theory in the exterior region due to the fact that the redshift effect at the event horizon acts as an unstable blueshift mechanism in the backwards direction in time. In the extremal case, however, the redshift effect degenerates and hence yields a much milder blueshift effect when viewed in the backwards direction. In this paper, we construct a definitive (bijective) non-degenerate scattering theory for the wave equation on extremal Reissner-Nordström backgrounds. We make use of physical-space energy norms which are non-degenerate both at the event horizon and at null infinity. As an application of our theory we present a construction of a large class of smooth, exponentially decaying modes. We also derive scattering results in the black hole interior region.
众所周知，由于事件视界处的红移效应在时间反方向上充当不稳定的蓝移机制，次极端黑洞背景在外部区域不允许存在（双射）非简并散射理论。然而，在极端情况下，红移效应退化，因此在时间反方向上观察时产生的蓝移效应要温和得多。在本文中，我们为极端雷斯纳 - 诺德斯特龙背景下的波动方程构建了一个确定的（双射）非简并散射理论。我们使用在事件视界和零无穷远处都非简并的物理空间能量范数。作为我们理论的一个应用，我们给出了一大类光滑、指数衰减模式的构造。我们还推导了黑洞内部区域的散射结果。
