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

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 $\mathcal{I}$ instead of ${\mathcal{C}}^k$ regularity at $\mathcal{I}$, allowing to characterise completely and without loss the classes of initial data ensuring a certain order of peeling at $\mathcal{I}$. 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.

arXiv comment51 pages

References in corpus (5)

• Existence of non-trivial, vacuum, asymptotically simple space-times
• Decay of Solutions of the Wave Equation in the Kerr Geometry
• Pointwise decay for the Maxwell field on black hole space-times
• Peeling of Dirac and Maxwell fields on a Schwarzschild background
• Conformal scattering for a nonlinear wave equation on a curved background

Cited by in corpus (9)

• Sharp decay estimates for massless Dirac fields on a Schwarzschild background
• Peeling of Dirac fields on Kerr spacetimes
• Hörmander's method for the characteristic Cauchy problem and conformal scattering for a non linear wave equation
• Conformal scattering theory for the linearized gravity fields on Schwarzschild spacetime
• Conformal scattering theory for the Dirac field on Kerr spacetime
• Peeling for tensorial wave equations on Schwarzschild spacetime
• Cauchy and Goursat problems for the generalized spin zero rest-mass fields on Minkowski spacetime
• Conformal scattering theories for tensorial wave equations on Schwarzschild spacetime
• Conformal Scattering of Maxwell Potentials