Sharp Decay for Teukolsky Equation in Kerr Spacetimes

Abstract

AbstractIn this work, we derive the global sharp decay, as both a lower and an upper bounds, for the spin $$\pm {\mathfrak {s}}$$ ± 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 $$({\mathfrak {s}}=0)$$ ( s = 0 ) , the Maxwell field $$({\mathfrak {s}}=1)$$ ( s = 1 ) and the linearized gravity $$({\mathfrak {s}}=2)$$ ( 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.