Time-Translation Invariance of Scattering Maps and Blue-Shift Instabilities on Kerr Black Hole Spacetimes

TL;DR

This paper investigates blue-shift instabilities in Kerr black hole spacetimes, demonstrating how solutions to the wave equation can have finite radiation fields but infinite local energy near horizons, using scattering theory and invariance properties.

Contribution

It provides a unified, elementary treatment of blue-shift and red-shift instabilities, revealing the role of scattering map invariance in these phenomena.

Findings

Solutions with vanishing radiation fields but infinite local energy near horizons.

Demonstration of time-translation invariance of scattering maps.

Explicit construction of solutions illustrating instabilities.

Abstract

In this paper, we provide an elementary, unified treatment of two distinct blue-shift instabilities for the scalar wave equation on a fixed Kerr black hole background: the celebrated blue-shift at the Cauchy horizon (familiar from the strong cosmic censorship conjecture) and the time-reversed red-shift at the event horizon (relevant in classical scattering theory). Our first theorem concerns the latter and constructs solutions to the wave equation on Kerr spacetimes such that the radiation field along the future event horizon vanishes and the radiation field along future null infinity decays at an arbitrarily fast polynomial rate, yet, the local energy of the solution is infinite near any point on the future event horizon. Our second theorem constructs solutions to the wave equation on rotating Kerr spacetimes such that the radiation field along the past event horizon (extended into the black hole) vanishes and the radiation field along past null infinity decays at an arbitrarily fast polynomial rate, yet, the local energy of the solution is infinite near any point on the Cauchy horizon. The results make essential use of the scattering theory developed in [M. Dafermos, I. Rodnianski and Y. Shlapentokh-Rothman, A scattering theory for the wave equation on Kerr black hole exteriors, preprint (2014) available at \url{http://arxiv.org/abs/1412.8379}] and exploit directly the time-translation invariance of the scattering map and the non-triviality of the transmission map.