Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
2009-08-28
Adv.Math.226:484-540,2011
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
43 pages, statement of Price's Law clarified and some minor corrections, to appear in Adv. Math
Scientific paper
10.1016/j.aim.2010.06.026
Price's Law states that linear perturbations of a Schwarzschild black hole fall off as $t^{-2\ell-3}$ for $t \to \infty$ provided the initial data decay sufficiently fast at spatial infinity. Moreover, if the perturbations are initially static (i.e., their time derivative is zero), then the decay is predicted to be $t^{-2\ell-4}$. We give a proof of $t^{-2\ell-2}$ decay for general data in the form of weighted $L^1$ to $L^\infty$ bounds for solutions of the Regge--Wheeler equation. For initially static perturbations we obtain $t^{-2\ell-3}$. The proof is based on an integral representation of the solution which follows from self--adjoint spectral theory. We apply two different perturbative arguments in order to construct the corresponding spectral measure and the decay bounds are obtained by appropriate oscillatory integral estimates.
Donninger Roland
Schlag Wilhelm
Soffer Avy
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