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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.