A proof of Price's Law on Schwarzschild black hole manifolds for all angular momenta

Price's Law states that linear perturbations of a Schwarzschild black hole fall off as t−2ℓ−3 for t→∞ 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−2ℓ−4. We give a proof of t−2ℓ−2 decay for general data in the form of weighted L1 to L∞ bounds for solutions of the Regge-Wheeler equation. For initially static perturbations we obtain t−2ℓ−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.