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

Miguel Duarte, Justin C. Feng, Edgar Gasperín, David Hilditch

TL;DR

The paper develops an adapted Hörmander asymptotic systems framework to the good-bad-ugly model and extends it to asymptotically flat spacetimes with stratified null forms, enabling polyhomogeneous, log-rich asymptotic expansions near null infinity. It shows that in Minkowski space the good field generically carries leading-order logarithms, with the bad field acquiring logs through nonlinear couplings, while the ugly field can avoid leading logs but may carry higher-order logarithms depending on the parameter $p$ and SNFs. Extending to the Einstein equations in generalized harmonic gauge, the authors derive a generalized good-bad-ugly system and construct polyhomogeneous expansions with logs up to order $3n-2$, using a new outgoing null vector to facilitate curved-spacetime integration. A key result is an obstruction to peeling in $oldsymbol{ extPsi}_2$ induced by leading logs, which persists despite gauge and constraint adjustments, though carefully chosen initial data or gluing can potentially eliminate good logs and restore peeling in some regimes. Overall, the work informs numerical relativity near null infinity by pinpointing where logarithmic terms arise and how gauge/initial-data choices influence asymptotic decay of the Weyl scalars.

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~$Ψ_n$ that is ensured by smoothness of null infinity. The leading order obstruction appears in~$Ψ_2$ and, in agreement with the literature, can only be suppressed by a careful choice of initial

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

TL;DR

The paper develops an adapted Hörmander asymptotic systems framework to the good-bad-ugly model and extends it to asymptotically flat spacetimes with stratified null forms, enabling polyhomogeneous, log-rich asymptotic expansions near null infinity. It shows that in Minkowski space the good field generically carries leading-order logarithms, with the bad field acquiring logs through nonlinear couplings, while the ugly field can avoid leading logs but may carry higher-order logarithms depending on the parameter and SNFs. Extending to the Einstein equations in generalized harmonic gauge, the authors derive a generalized good-bad-ugly system and construct polyhomogeneous expansions with logs up to order , using a new outgoing null vector to facilitate curved-spacetime integration. A key result is an obstruction to peeling in induced by leading logs, which persists despite gauge and constraint adjustments, though carefully chosen initial data or gluing can potentially eliminate good logs and restore peeling in some regimes. Overall, the work informs numerical relativity near null infinity by pinpointing where logarithmic terms arise and how gauge/initial-data choices influence asymptotic decay of the Weyl scalars.

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~ of the Weyl scalars, where~ 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~ and, in agreement with the literature, can only be suppressed by a careful choice of initial

Paper Structure

This paper contains 26 sections, 2 theorems, 134 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Geometric setup
  3. Representation of the metric
  4. 3+1 decomposition
  5. Wave operators
  6. Model equations
  7. Assumptions
  8. Minkowski spacetime
  9. Good field
  10. On the generality of the first order log
  11. Bad field
  12. Ugly field
  13. The influence of $p$
  14. Good-bad-ugly system
  15. Asymptotically flat spacetimes
  16. ...and 11 more sections

Key Result

Theorem 1

Let $X^{\ul{\alpha}}=(T,X^{\ul{i}})$ be an asymptotically Cartesian coordinate system with an associated covariant derivative $\mathring{\nabla}$ and let $\mathcal{S}$ be a Cauchy hypersurface defined by the condition $T=0$. The good-bad-ugly system defined as, where $\mathring{\square}:=\eta^{ab}\mathring{\nabla}_a\mathring{\nabla}_b$ and $\eta$ is the Minkowski metric, admits formal polyhomogen

Figures (1)

  • Figure 1: A schematic of our geometric setup. The method proceeds first by integrating out along $c$, an integral curve of the outgoing null-vector $\psi^a$ and then up along integral curves of $\partial_T^a$.

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2