# The Weak Null Condition in Free-evolution Schemes for Numerical Relativity: Dual Foliation GHG with Constraint Damping

## TL;DR

This paper develops a method to ensure optimal decay rates of the metric near null-infinity in numerical relativity, enabling explicit treatment of future null-infinity by modifying field equations with constraints.

## Contribution

It generalizes asymptotic expansion techniques for nonlinear wave equations to hyperbolic systems and demonstrates how to modify equations for optimal decay rates even with constraint violations.

## Key findings

- Optimal decay rates achieved with constraint modifications
- Bondi mass loss formula recovered with small constraint violations
- Divergent terms in mass can be replaced using constraints and Einstein equations

## Abstract

All strategies for the treatment of future null-infinity in numerical relativity involve some form of regularization of the field equations. In a recent proposal that relies on the dual foliation formalism this is achieved by the use of an asymptotically Minkowskian generalized harmonic tensor basis. For the scheme to work however, derivatives of certain coordinate light-speeds must decay fast enough. Presently, we generalize the method of asymptotic expansions for nonlinear wave equations to treat first order symmetric hyperbolic systems. We then use this heuristic tool to extract the expected rates of decay of the metric near null-infinity in a free-evolution setting. We show, within the asymptotic expansion, that by carefully modifying the non-principal part of the field equations by the addition of constraints, we are able to obtain optimal decay rates even when the constraints are violated. The light-speed condition can hence be satisfied, which paves the way for the explicit numerical treatment of future null-infinity. We then study the behavior of the Trautman-Bondi mass under the decay results predicted by the asymptotic expansion. Naively the mass seems to be unbounded, but we see first that the divergent terms can be replaced with a combination of the constraints and the Einstein field equations, and second that the Bondi mass loss formula is recovered within the framework. Both of the latter results hold in the presence of small constraint violations.