High Order Asymptotic Expansions of a Good-Bad-Ugly Wave Equation
arXiv:2101.07068 · doi:10.1088/1361-6382/abfed2
Abstract
A heuristic method to find asymptotic solutions to a system of non-linear wave equations near null infinity is proposed. The non-linearities in this model, dubbed good-bad-ugly, are known to mimic the ones present in the Einstein field equations (EFE) and we expect to be able to exploit this method to derive an asymptotic expansion for the metric in General Relativity (GR) close to null infinity that goes beyond first order as performed by Lindblad and Rodnianski for the leading asymptotics. For the good-bad-ugly model, we derive formal expansions in which terms proportional to the logarithm of the radial coordinate appear at every order in the bad field, from the second order onward in the ugly field but never in the good field. The model is generalized to wave operators built from an asymptotically flat metric and it is shown that it admits polyhomogeneous asymptotic solutions. Finally we define stratified null forms, a generalization of standard null forms, which capture the behavior of different types of field, and demonstrate that the addition of such terms to the original system bears no qualitative influence on the type of asymptotic solutions found.
arXiv comment15 pages, 1 figure
References in corpus (12)
- The Asymptotic Behavior of Massless Fields and the Memory Effect
- A Scheme to Numerically Evolve Data for the Conformal Einstein Equation
- Regularity of the Einstein Equations at Future Null Infinity
- From Now to Timelike Infinity on a Finite Grid
- Tetrad formalism for numerical relativity on conformally compactified constant mean curvature hypersurfaces
- Spherical symmetry as a test case for unconstrained hyperboloidal evolution
- Black hole initial data on hyperboloidal slices
- Polyhomogeneous expansions from time symmetric initial data
- The Weak Null Condition in Free-evolution Schemes for Numerical Relativity: Dual Foliation GHG with Constraint Damping
- Unconstrained hyperboloidal evolution of black holes in spherical symmetry with GBSSN and Z4c
- A conformal approach to numerical calculations of asymptotically flat spacetimes
- Free evolution of the hyperboloidal initial value problem in spherical symmetry