Sharp decay for Teukolsky equation in Kerr spacetimes

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.

The published version

References in corpus (5)

• Semilinear wave equations on the Schwarzschild manifold I: Local decay estimates
• The Case Against Smooth Null Infinity III: Early-Time Asymptotics for Higher $\ell$-Modes of Linear Waves on a Schwarzschild Background
• Nonlinear radiation gauge for near Kerr spacetimes
• Pointwise decay for the wave equation on nonstationary spacetimes
• Pointwise decay for semilinear wave equations on Kerr spacetimes