Late-time tails and mode coupling of linear waves on Kerr spacetimes

Abstract

We provide a rigorous derivation of the precise late-time asymptotics for solutions to the scalar wave equation on subextremal Kerr backgrounds, including the asymptotics for projections to angular frequencies ℓ≥1 and ℓ≥2. The ℓ-dependent asymptotics on Kerr spacetimes differ significantly from the non-rotating Schwarzschild setting (“Price's law”). The main differences with Schwarzschild are slower decay rates for higher angular frequencies and oscillations along the null generators of the event horizon. We introduce a physical space-based method that resolves the following two main difficulties for establishing ℓ-dependent asymptotics in the Kerr setting: 1) the coupling of angular modes and 2) a loss of ellipticity in the ergoregion. Our mechanism identifies and exploits the existence of conserved charges along null infinity via a time invertibility theory, which in turn relies on new elliptic estimates in the full black hole exterior. This framework is suitable for addressing conflicting results on the numerology of Kerr late-time asymptotics appearing in the numerics literature and definitively determining the correct numerology.