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

Author(s): Dafermos, Mihalis; Shlapentokh-Rothman, Yakov

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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 Dafermos, Rodnianski and Shlapentokh-Rothman (A scattering theory for the wave equation on Kerr black hole exteriors (2014). arXiv:1412.8379) and exploit directly the time-translation invariance of the scattering map and the non-triviality of the transmission map.
Publication Date: Mar-2017
Electronic Publication Date: 18-Oct-2016
Citation: Dafermos, Mihalis, Shlapentokh-Rothman, Yakov. (2017). Time-Translation Invariance of Scattering Maps and Blue-Shift Instabilities on Kerr Black Hole Spacetimes. COMMUNICATIONS IN MATHEMATICAL PHYSICS, 350 (985 - 1016. doi:10.1007/s00220-016-2771-z
DOI: doi:10.1007/s00220-016-2771-z
ISSN: 0010-3616
EISSN: 1432-0916
Pages: 985 - 1016
Type of Material: Journal Article
Journal/Proceeding Title: COMMUNICATIONS IN MATHEMATICAL PHYSICS
Version: Author's manuscript



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