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

TL;DR

This paper proves Price's Law decay rates for linear perturbations of Schwarzschild black holes across all angular momenta, using spectral theory and oscillatory integral estimates.

Contribution

It establishes decay rates for all angular momenta in Price's Law for Schwarzschild black holes, including static initial data, via spectral analysis and integral estimates.

Findings

Proves $t^{-2 ext{ extellipsis}}$ decay for general data

Establishes $t^{-2 ext{ extellipsis}}$ decay for static initial data

Uses spectral theory and oscillatory integrals for decay bounds

Abstract

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.