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

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.

arXiv comment71 pages, 13 figures

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• The Case Against Smooth Null Infinity I: Heuristics and Counter-Examples
• Quasinormal modes in extremal Reissner-Nordström spacetimes
• Strong Cosmic Censorship in the presence of matter: the decisive effect of horizon oscillations on the black hole interior geometry
• Scattering from Infinity of the Maxwell Klein Gordon Equations in Lorenz Gauge
• Conformal scattering theory for the linearized gravity fields on Schwarzschild spacetime
• The Case Against Smooth Null Infinity V: Early-Time Asymptotics of Linearised Gravity Around Schwarzschild for Fixed Spherical Harmonic Modes
• Peeling at extreme black hole horizons
• Conformal Scattering of Maxwell Potentials
• Conformal scattering theories for tensorial wave equations on Schwarzschild spacetime