Regularity at space-like and null infinity

Abstract

We extend Penrose's peeling model for the asymptotic behaviour of solutions to the scalar wave equation at null infinity on asymptotically flat backgrounds, which is well understood for flat space-time, to Schwarzschild and the asymptotically simple space-times of Corvino-Schoen/Chrusciel-Delay. We combine conformal techniques and vector field methods: a naive adaptation of the 'Morawetz vector field' to a conformal rescaling of the Schwarzschild metric yields a complete scattering theory on Corvino-Schoen/Chrusciel-Delay space-times. A good classification of solutions that peel arises from the use of a null vector field that is transverse to null infinity to raise the regularity in the estimates. We obtain a new characterization of solutions admitting a peeling at a given order that is valid for both Schwarzschild and Minkowski space-times. On flat space-time, this allows larger classes of solutions than the characterizations used since Penrose's work. Our results establish the validity of the peeling model at all orders for the scalar wave equation on the Schwarzschild metric and on the corresponding Corvino-Schoen/Chrusciel-Delay space-times.

Peeling of Dirac and Maxwell fields on a Schwarzschild background

“…The question of regularity of null infinity and asymptotic simplicity has now been resolved in various ways, Christodoulou-Klainerman [2], Corvino [5], Chrusciel and Delay [3,4], Corvino-Schoen [6], Friedrich (see [10] for a survey of his contributions) and Klainerman-Nicolò [12,13,14]. However, even in the simple case of the Schwarzschild metric, it was not at all clear, until the authors provided a first element of answer in [16], whether zero rest-mass fields admit peeling properties for reasonably large classes of initial data.…”

The Conformal Approach to Asymptotic Analysis

“…This is expected to be generic since a physically relevent asymptotically flat spacetime ought to be a short-range perturbation of a Schwarzschild spacetime. The question was however answered in the affirmative in [27,28] by treating the Schwarzschild case. The key idea was to reformulate the peeling property in terms of energy estimates instead of pointwise behaviour along outgoing null geodesics.…”

“…We could simply commute T into the equation ; since it is Killing, we would immediately obtain identitites similar to (28) for T kφ . Although this would again be a perfectly valid definition, it would not extend naturally to other spacetimes such as Schwarzschild, because in these spacetimes, the Killing form of T induces terms that are delicate to handle in the higher order estimates (see [27] for details). Another vector field that we can use is ∂ R .…”

“…The details can be found in [27] for the wave equation and [28] for Dirac and Maxwell fields. We now express the definition of the peeling at any order on the Schwarzschild metric and its characterization in terms of classes of initial data.…”

“…The essential reason for this is that the norms in the Schwarzschild situation are uniform in the mass m of the black hole on any given bounded interval ]0, M ]. The details can be found in [27,28].…”