Stability for linearized gravity on the Kerr spacetime

TL;DR

This paper establishes decay estimates for solutions to the linearized Einstein equations on Kerr black holes, advancing the understanding of their stability, especially in the slowly rotating case, and linking it to Teukolsky equation estimates.

Contribution

It provides the first stability results for linearized gravity on Kerr spacetime in the slowly rotating case, connecting stability to Teukolsky equation estimates across the full subextreme range.

Findings

Proved integrated energy and pointwise decay estimates for linearized gravity on Kerr.

Reduced the linearized stability problem to Teukolsky equation estimates.

Established the first stability results for Kerr in the slowly rotating case.

Abstract

In this paper we prove integrated energy and pointwise decay estimates for solutions of the vacuum linearized Einstein equation on the domain of outer communication of the Kerr black hole spacetime. The estimates are valid for the full subextreme range of Kerr black holes, provided integrated energy estimates for the Teukolsky equation hold. For slowly rotating Kerr backgrounds, such estimates are known to hold, due to the work of one of the authors. The results in this paper thus provide the first stability results for linearized gravity on the Kerr background, in the slowly rotating case, and reduce the linearized stability problem for the full subextreme range to proving integrated energy estimates for the Teukolsky equation. This constitutes an essential step towards a proof of the black hole stability conjecture, i.e. the statement that the Kerr family is dynamically stable, one of the central open problems in general relativity.