Peeling of Dirac field on Kerr spacetime

Abstract

In a recent paper with Nicolas [Ann. Henri Poincaré 20(10), 3419–3470 (2019); arXiv:1801.08996], 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 Mason and Nicolas [J. Inst. Math. Jussieu 8(1), 179–208 (2009); arXiv:gr-qc/0701049 and L. Mason and J.-P. Nicolas, J. Geom. Phys. 62(4), 867–889 (2012); arXiv:1101.4333] 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 peeling at all orders in terms of Sobolev regularity near I, instead of Ck regularity at I, and then provides the optimal spaces of initial data such that the associated solution satisfies peeling at a given order. The results confirm that the analogous decay and regularity assumptions on initial data in the Minkowski and Kerr spacetimes 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.