Height-function-based 4D reference metrics for hyperboloidal evolution

TL;DR

This paper investigates how different height-function-based reference metrics influence hyperboloidal evolution in numerical relativity, focusing on stability and asymptotic properties in spherical symmetry, with implications for future 3D simulations.

Contribution

It introduces and compares ten height-function-based reference metrics for hyperboloidal slices, including the first non-linear Einstein evolutions of hyperboloidal layers.

Findings

Identified features of effective reference metrics for hyperboloidal evolution.

Demonstrated long-term numerical stability with various reference metrics.

First non-linear Einstein evolutions of hyperboloidal layers in spherical symmetry.

Abstract

Hyperboloidal slices are spacelike slices that reach future null infinity. Their asymptotic behaviour is different from Cauchy slices, which are traditionally used in numerical relativity simulations. This work uses free evolution of the formally-singular conformally compactified Einstein equations in spherical symmetry. One way to construct gauge conditions suitable for this approach relies on building the gauge source functions from a time-independent background spacetime metric. This background reference metric is set using the height function approach to provide the correct asymptotics of hyperboloidal slices of Minkowski spacetime. The present objective is to study the effect of different choices of height function on hyperboloidal evolutions via the reference metrics used in the gauge conditions. A total of 10 reference metrics for Minkowski are explored, identifying some of their desired features. They include 3 hyperboloidal layer constructions, evolved with the non-linear Einstein equations for the first time. Focus is put on long-term numerical stability of the evolutions, including small initial gauge perturbations. The results will be relevant for future (puncture-type) hyperboloidal evolutions, 3D simulations and the development of coinciding Cauchy and hyperboloidal data, among other applications.