Peeling of Dirac fields on Kerr spacetimes

TL;DR

This paper extends the analysis of peeling properties from scalar fields to Dirac fields on Kerr spacetimes, using conformal compactification and energy estimates to characterize decay and regularity at null infinity.

Contribution

It introduces a novel approach to define peeling for Dirac fields on Kerr backgrounds, generalizing previous scalar field results to spinor fields with optimal initial data spaces.

Findings

Peeling behavior for Dirac fields is established on Kerr spacetimes.

Decay and regularity assumptions match those in Minkowski space.

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

Abstract

In a recent paper with J.-P. Nicolas [J.-P. Nicolas and P.T. Xuan, Annales Henri Poincare 2019], we studied the peeling for scalar fields on Kerr metrics. The present work extends these results to Dirac fields on the same geometrical background. We follow the approach initiated by L.J. Mason and J.-P. Nicolas [L. Mason and J.-P. Nicolas, J.Inst.Math.Jussieu 2009; L. Mason and J.-P. Nicolas, J.Geom.Phys 2012] on the Schwarzschild spacetime and extended to Kerr metrics for scalar fields. The method combines the Penrose conformal compactification and geometric energy estimates in order to work out a definition of the peeling at all orders in terms of Sobolev regularity near $\mathscr{I}$, instead of ${\mathcal C}^k$ regularity at $\mathscr{I}$, then provides the optimal spaces of initial data such that the associated solution satisfies the peeling at a given order. The results confirm that the analogous decay and regularity assumptions on initial data in Minkowski and in Kerr produce the same regularity across null infinity. 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.