2013/05/08 by Semyon Dyatlov
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Asymptotic expansion #Black Holes and Theoretical Physics #Bounded function #Exponential decay #Exponential function #Geometric Analysis and Curvature Flows #Integrable system #Lax pair #Norm (philosophy) #Projector #gr-qc #math.AP #math.SP
paper · pdf · doi:10.1007/s00220-014-2255-y
45 pages, 4 figures
arxiv created 2013/05/08 · openalex publication_date 2015/01/28 · arxiv updated 2015/02/02 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We apply the results of arXiv:1301.5633 to describe asymptotic behavior of linear waves on stationary Lorentzian metrics with r-normally hyperbolic trapped sets, in particular Kerr and Kerr-de Sitter metrics with |a|<M and MΛa << 1. We prove that if the initial data is localized at frequencies λ>> 1, then the energy norm of the solution is bounded by O(λ1/2 exp(-(νmin - ε)t/2) + λ^(-∞)), for t < C logλ, where νmin 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(λexp(-(νmin -ε)t) + λ^(-∞)) remainder; this splitting can be viewed as an analog of resonance expansion. Moreover, for the Kerr-de Sitter case we study quasi-normal modes; under a dynamical pinching condition, a Weyl law in a band holds.