Asymptotics of Linear Waves and Resonances with Applications to Black Holes

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.