Asymptotics of Linear Waves and Resonances with Applications to Black Holes
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220_2014_Article_2255.pdf
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Author(s)
Dyatlov, Semen
Date Issued
January 2015
Journal
Communications in Mathematical Physics
Publisher
Springer Berlin Heidelberg
Citation
Dyatlov, Semyon. “Asymptotics of Linear Waves and Resonances with Applications to Black Holes.” Communications in Mathematical Physics 335.3 (2015): 1445–1485.
Version
Author's final manuscript
Abstract
We describe asymptotic behavior of linear waves on Kerr(–de Sitter) black holes and more general Lorentzian manifolds, providing a quantitative analysis of the ringdown phenomenon. In particular we prove that if the initial data is localized at frequencies ∼λ≫1, then the energy norm of the solution is bounded by O(λ[superscript 1/2]e[superscript -(νmin−ε)t/2]) + O(λ[superscript−∞]), for t ≤ C log λ,where ν[subscript min] is a natural dynamical quantity. The key tool is a microlocal projector splitting the solution into a component with controlled rate of exponential decay and an O(λe[superscript −(νmin−ε)t]) + O(λ[superscript−∞]) remainder. This splitting generalizes expansions into quasi-normal modes available in completely integrable settings. In the case of generalized Kerr(–de Sitter) black holes satisfying certain natural conditions, quasi-normal modes are localized in bands and satisfy a precise counting law.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1007/s00220-014-2255-y
