Mode stability for the Teukolsky equation on extremal and subextremal Kerr spacetimes

TL;DR

This paper proves mode stability for the Teukolsky equation on both extremal and subextremal Kerr black holes, showing no exponentially growing or real-axis modes, and provides bounds on scattering coefficients.

Contribution

It extends mode stability results to extremal Kerr spacetimes, a previously unresolved case, and refines quantitative bounds on scattering coefficients.

Findings

No exponentially growing modes on Kerr spacetimes.

Bounded transmission and reflection coefficients in frequency space.

First mode stability results for extremal Kerr outside axisymmetry.

Abstract

We prove that there are no exponentially growing modes nor modes on the real axis for the Teukolsky equation on Kerr black hole spacetimes, both in the extremal and subextremal case. We also give a quantitative refinement of mode stability. As an immediate application, we show that the transmission and reflection coefficients of the scattering problem are bounded, independently of the specific angular momentum of the black hole, in any compact set of real frequencies excluding zero frequency and the superradiant threshold. While in the subextremal setting these results were known previously, the extremal case is more involved and has remained an open problem. Ours are the first results outside axisymmetry and could serve as a preliminary step towards understanding boundedness, scattering and decay properties of general solutions to the Teukolsky equation on extremal Kerr black holes.