A non-degenerate scattering theory for the wave equation on extremal Reissner-Nordstrom

TL;DR

This paper develops a new non-degenerate scattering theory for the wave equation on extremal Reissner-Nordstrom black holes, overcoming previous limitations caused by the redshift effect's degeneracy.

Contribution

It introduces a bijective scattering framework using physical-space energy norms that are non-degenerate at the horizon and infinity, specifically for extremal black holes.

Findings

Constructed a bijective scattering theory for extremal Reissner-Nordstrom backgrounds.

Built a large class of smooth exponentially decaying modes.

Derived scattering results within the black hole interior.

Abstract

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 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-Nordstrom 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.