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Asymptotics of Linear Waves and Resonances with Applications to Black Holes

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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 \({\sim \lambda \gg 1}\), then the energy norm of the solution is bounded by \({\mathcal{O}(\lambda^{1/2} e^{-(\nu_{\min}-\varepsilon)t/2}) + \mathcal{O}(\lambda^{-\infty}), \,\,{\rm for}\,\, t \leq C \log \lambda, \,\,{\rm where}\,\, \nu_{\min}}\) 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 \({\mathcal{O}(\lambda e^{-(\nu_{\min}-\varepsilon)t}) + \mathcal{O}(\lambda^{-\infty})}\) 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.

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  1. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA, 02139, USA

    Semyon Dyatlov

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Communicated by P. T. Chruściel

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Dyatlov, S. Asymptotics of Linear Waves and Resonances with Applications to Black Holes. Commun. Math. Phys. 335, 1445–1485 (2015). https://doi.org/10.1007/s00220-014-2255-y

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