Uniform energy bound and Morawetz estimate for extreme components of spin fields in the exterior of a slowly rotating Kerr black hole II: linearized gravity

TL;DR

This paper establishes energy bounds and Morawetz estimates for extreme spin components of linearized gravity in the exterior of a slowly rotating Kerr black hole, advancing the understanding of black hole stability.

Contribution

It introduces a new approach to derive energy and Morawetz estimates for spin ±2 components by analyzing a coupled wave system derived from the Teukolsky equation.

Findings

Proved energy and Morawetz estimates for spin ±2 components.

Developed a linear spin-weighted wave system from the Teukolsky equation.

Paves the way for proving linear stability of slowly rotating Kerr black holes.

Abstract

This second part of the series treats spin $\pm2$ components (or extreme components), that satisfy the Teukolsky master equation, of the linearized gravity in the exterior of a slowly rotating Kerr black hole. For each of these two components, after performing a first-order differential operator once and twice, the resulting equations together with the Teukolsky master equation itself constitute a linear spin-weighted wave system. An energy and Morawetz estimate for spin $\pm 2$ components is proved by treating this system. This is a first step in a joint work (Andersson et al. in Stability for linearized gravity on the Kerr spacetime, arXiv:1903.03859, 2019) in addressing the linear stability of slowly rotating Kerr metrics.