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

TL;DR

This paper rigorously derives the late-time decay rates of scalar waves on Kerr black holes, revealing differences from Schwarzschild due to rotation, and introduces a new method to handle mode coupling and ergoregion issues.

Contribution

It provides the first rigorous derivation of ll-dependent late-time asymptotics for scalar waves on Kerr backgrounds, addressing mode coupling and ergoregion challenges.

Findings

ll-dependent decay rates differ from Schwarzschild

Slower decay for higher angular frequencies

Oscillations along the event horizon null generators

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 $\ell\geq 1$ and $\ell\geq 2$. The $\ell$-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 $\ell$-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 resolving the conflicting numerology in Kerr late-time asymptotics that appears in the numerics literature.