An asymptotic systems approach for the good–bad–ugly model with application to general relativity

Abstract

We employ an adapted version of Hörmander’s asymptotic systems method to show heuristically that the standard good–bad–ugly model admits formal polyhomogeneous asymptotic solutions near null infinity. In a related earlier approach, our heuristics were unable to capture potential leading order logarithmic terms appearing in the asymptotic solution of the good equation (the standard wave equation). Presently, we work with an improved method which overcomes this shortcoming, allowing the faithful treatment of a larger class of initial data in which such logarithmic terms are manifest. We then generalize this method to encompass models that include stratified null forms as sources and whose wave operators are built from an asymptotically flat metric. We then apply this result to the Einstein field equations in generalized harmonic gauge and compute the leading decay in R−1 of the Weyl scalars, where R is a suitably defined radial coordinate. We detect an obstruction to peeling, a decay statement on the Weyl scalars that is ensured by smoothness of null infinity. The leading order obstruction appears in Ψ2 and, in agreement with the literature, can only be suppressed by a careful choice of initial data.