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

TL;DR

This work proves ℓ–dependent Price‑Law decay for linear Regge–Wheeler perturbations on Schwarzschild black holes by constructing a detailed spectral representation via Jost solutions and the Green’s function. The authors build precise small‑ and large‑energy perturbative solutions, establish the spectral measure at zero energy, and derive weighted L1→L∞ decay bounds for the wave evolution, obtaining $t^{-2\ell-2}$ decay for general data and $t^{-2\ell-3}$ for initially static data. A key technical achievement is the intricate control of oscillatory integrals across energy regimes, including sharp zero‑energy asymptotics and resonance absence for the physically relevant cases. The results illuminate angular‑momentum–dependent decay and set the stage for summing over ℓ in a companion semiclassical analysis (DSS2), with implications for black hole stability and gravitational wave modeling.

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.

Theorems & Definitions (82)

• Lemma 2.1
• proof
• Corollary 2.1
• proof
• Theorem 2.1
• Remark 2.1
• Remark 2.2
• Lemma 3.1
• proof
• Lemma 3.2