Asymptotics of linear waves and resonances with applications to black holes

TL;DR

This paper analyzes the asymptotic behavior of linear waves and resonances in black hole spacetimes, providing bounds on energy decay and studying quasi-normal modes under certain conditions.

Contribution

It extends previous results to Kerr and Kerr-de Sitter metrics, introducing a microlocal splitting method and establishing a Weyl law for quasi-normal modes.

Findings

Energy norm bounded by O(λ^{1/2} exp(-( u_min - ε)t/2)) for high-frequency initial data

Microlocal projector effectively separates solution components with controlled decay rates

Weyl law for quasi-normal modes under dynamical pinching condition

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 M\Lambda a << 1. We prove that if the initial data is localized at frequencies \lambda >> 1, then the energy norm of the solution is bounded by O(\lambda^{1/2} exp(-(\nu_min - \epsilon)t/2) + \lambda^(-\infty)), for t < C log\lambda, where \nu_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(\lambda exp(-(\nu_min -\epsilon)t) + \lambda^(-\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.