Sharp decay for Teukolsky equation in Kerr spacetimes

TL;DR

This paper establishes sharp decay rates for solutions to the Teukolsky equation in Kerr spacetimes, confirming Price's law and introducing a novel conservation law approach applicable to various fields.

Contribution

It derives the first global sharp decay estimates for Teukolsky equation solutions in Kerr spacetimes, using a new conservation law method.

Findings

Confirmed Price's law decay conjecture.

Established decay bounds for scalar, Maxwell, and linearized gravity fields.

Introduced a novel conservation law approach for asymptotic analysis.

Abstract

In this work, we derive the global sharp decay, as both a lower and an upper bounds, for the spin $\pm \mathfrak{s}$ components, which are solutions to the Teukolsky equation, in the black hole exterior and on the event horizon of a slowly rotating Kerr spacetime. These estimates are generalized to any subextreme Kerr background under an integrated local energy decay estimate. Our results apply to the scalar field $(\mathfrak{s}=0)$, the Maxwell field $(\mathfrak{s}=1)$ and the linearized gravity $(\mathfrak{s}=2)$ and confirm the Price's law decay that is conjectured to be sharp. Our analyses rely on a novel global conservation law for the Teukolsky equation, and this new approach can be applied to derive the precise asymptotics for solutions to semilinear wave equations.