Boundedness and decay for the Teukolsky equation on Kerr spacetimes I: the case $|a|\ll M$

TL;DR

This paper establishes boundedness and decay of solutions to the Teukolsky equation on Kerr spacetimes with small angular momentum, advancing the understanding of gravitational perturbations and stability.

Contribution

It introduces generalized higher order quantities extending previous Schwarzschild results, crucial for proving linear stability of Kerr black holes.

Findings

Proved polynomial decay of solutions for small angular momentum Kerr.

Extended the transformation theory to Kerr spacetime.

Provided foundational bounds for future stability proofs.

Abstract

We prove boundedness and polynomial decay statements for solutions of the spin $\pm2$ Teukolsky equation on a Kerr exterior background with parameters satisfying $|a|\ll M$. The bounds are obtained by introducing generalisations of the higher order quantities $P$ and $\underline{P}$ used in our previous work on the linear stability of Schwarzschild. The existence of these quantities in the Schwarzschild case is related to the transformation theory of Chandrasekhar. In a followup paper, we shall extend this result to the general sub-extremal range of parameters $|a|<M$. As in the Schwarzschild case, these bounds provide the first step in proving the full linear stability of the Kerr metric to gravitational perturbations.