paper

Asymptotics of linear waves and resonances with applications to black holes

arXiv:1305.1723 · doi:10.1007/s00220-014-2255-y

Abstract

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(λ^{1/2} exp(-(ν_min - ε)t/2) + λ^(-\infty)), for t < C logλ, where ν_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(λexp(-(ν_min -ε)t) + λ^(-\infty)) 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.

45 pages, 4 figures

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