The good-bad-ugly system near spatial infinity on flat spacetime

Abstract

A model system of equations that serves as a model for the Einstein field equation in generalised harmonic gauge called the good-bad-ugly system is studied in the region close to null and spatial infinity in Minkowski spacetime. This analysis is performed using H. Friedrich’s cylinder construction at spatial infinity and defining suitable conformally rescaled fields. The results are translated to the physical set up to investigate the relation between the polyhomogeneous expansions arising from the analysis of linear fields using the i0-cylinder framework and those obtained through a heuristic method based on Hörmander’s asymptotic system.

Polyhomogeneous spin-0 fields in Minkowski space–time

“…This review article gives an abridged discussion of the behaviour close to spatial and null infinity of a system of equations called the GBU model in Minkowski space-time based on [35,36]. The GBU system is a simple model that mimics some of the nonlinearities appearing in a recent reformulation of the Einstein field equations in generalized harmonic gauge well adapted to the hyperboloidal initial value problem using the dual-foliation formalism for numerical relativity (HypDF)-see [27,29,30].…”

“…In principle it could be possible to investigate the condition controlling appearance of the logarithmic terms in the bad field at the level of the initial data at each order in p. However, one would obtain unrevealing and convoluted expressions. This analysis was out of the scope of Duarte et al [35] and hence that of this review. This situation is in stark contrast to the solution for the good field where the appearance of the logarithmic terms can be cleanly controlled at the level of initial data through the coefficient D p,p,m .…”

“…Later, a method to recover all the missed higher-order logarithmic terms (in flat space-time) was obtained as a subproduct of the analysis of Gasperín & Pinto [36] for the calculation of the Newman-Penrose constants of spin-0 fields. In this review article, the discussion of the GBU-system close to spatial and null infinity of Duarte et al [35] and the higher order asymptotic system approach trough conservation laws in flat space-time discussed in [36] are revisited and presented in a coherent and contiguous way.…”

“…Interestingly, the failure of the classical peeling result obtained in [34] looks similar to that of Gasperín & Valiente Kroon [15] which uses a contrastingly different formal expansion scheme based on the CEFEs and conformal methods. To clarify the relation between these two results, in [35] the conformal method was applied to a model system of equations which is called the good–bad–ugly (GBU), upon which the bare-bones regularization of the HypDF approach is based—see [30]. The basic conclusion from Duarte et al [35] is that, in the simplest case of the GBU-system in Minkowski space–time, the logarithmic terms in [33] (and hence those of Duarte et al [34]) do not correspond to those of Gasperín & Valiente Kroon [15].…”

3D Evolution of a Good-Bad-Ugly-F Model on Compactified Hyperboloidal Slices

“…An interesting point that we have made no attempt whatsoever to understand here is the effect of choosing initial data that decay only very slowly towards I + . This would make direct contact with [43], where such initial data were considered for a subsector of the GBUF model. In the context of GR, such data has relevance to the question of the peeling property at I + (see [44][45][46][47] for recent work in this direction).…”