Peeling on Kerr spacetime~:linear and non linear scalar fields

TL;DR

This paper extends the analysis of peeling properties of scalar fields from Schwarzschild to Kerr spacetime, including nonlinear equations, using Sobolev regularity and energy estimates, applicable to all Kerr parameters.

Contribution

It introduces a Sobolev-based framework for peeling in Kerr spacetime, including nonlinear scalar fields, generalizing previous Schwarzschild results and confirming optimality.

Findings

Peeling characterized by Sobolev regularity near null infinity.

Results valid for all Kerr angular momenta, including fast rotation.

First analysis of nonlinear scalar fields in Kerr peeling context.

Abstract

We study the peeling on Kerr spacetime for fields satisfying conformally invariant linear and nonlinear scalar wave equations. We follow an approach initiated by L.J. Mason and the first author for the Schwarzschild metric, based on a Penrose compactification and energy estimates. This approach provides a definition of the peeling at all orders in terms of Sobolev regularity near ${\mathscr I}$ instead of ${\cal C}^k$ regularity at ${\mathscr I}$, allowing to characterise completely and without loss the classes of initial data ensuring a certain order of peeling at ${\mathscr I}$. This paper extends the construction to the Kerr metric, confirms the validity and optimality of the flat spacetime model (in the sense that the same regularity and fall-off assumptions on the data guarantee the peeling behaviour in flat spacetime and on the Kerr metric) and does so for the first time for a nonlinear equation. Our results are local near spacelike infinity and are valid for all values of the angular momentum of the spacetime, including for fast Kerr metrics.