Peeling on Kerr Spacetime: Linear and Semi-linear Scalar Fields

Abstract

We study the peeling on Kerr spacetime for fields satisfying conformally invariant linear and nonlinear scalar wave equations. We follow the approach initiated by L.J. Mason and the first author [23,24] 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 I instead of C k regularity at I , allowing to characterise completely and without loss the classes of initial data ensuring a certain order of peeling at 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.

Peeling of Dirac field on Kerr spacetime

“…Therefore, since (40) and (41) and by using again the generalized result of Hörmander (Appendix 5.1) with the zero initial data on the null boundary H + ∪ I + Q , we get Ξ A = 0 and then Ξ A = ∇ AA φ A = 0. So the solution of the system (38) is a solution of the system (35). For convenience, we denote by φ 1…”

“…In this paper and the recent work [32], we study the peeling on Kerr spacetime. The previous work [32] was an extension of the results of [26] to the Kerr metric and also treated the case of a semi-linear conformal wave equation. Here, we extend the results of [25] for Dirac fields to Kerr metrics.…”

“…Here, we extend the results of [25] for Dirac fields to Kerr metrics. The method is adapted from [25,26,32]. On the compactified Kerr spacetime, we choose a neighbourhood of spacelike infinity that is globally hyperbolic in order to ensure that the massless Dirac equation (the Weyl equation) has a well-posed Cauchy problem.…”

“…The paper is organised as follows: in Section 2, we recall the definition of the Kerr metric, construct its conformal compactification, define the neighbourhood Ω + * t 0 of spacelike infinity on which we shall perform our estimates as well as its foliation by spacelike slices H s (which are described in detail in [32]), present our choice of Newman-Penrose tetrad and calculate the associated spin coefficients in both the original spacetime and the rescaled one. Section 3 gives the detailed expression of the Weyl equation, some discussions on the Cauchy problem in the neighbourhood Ω + * t 0 ; we also introduce the conserved current, give the associated energy equality between the future and past boundaries of Ω + * t 0 and then we obtain simplified equivalent expressions of the energies on the slices of the foliation {H s } s .…”

Hörmander’s method for the characteristic Cauchy problem and conformal scattering for a nonlinear wave equation

“…In the past years, this approach by conformal techniques to understand the asymptotic behaviour of solutions of field equations in general relativity has been studied in various contexts: Mason and Nicolas [MN04] obtained the first analytic result for linear fields, followed by a peeling result on the Schwarzschild background, for scalar wave [MN09], spin 1/2 and spin 1 fields [MN12]. This has been later extended to non-linear waves on Kerr black holes [NP18]. As mentioned before, the conformal scattering construction was extended to a non-linear wave equation [Jou12].…”