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.