Towards a Stable Numerical Evolution of Strongly Gravitating Systems in General Relativity: The Conformal Treatments

TL;DR

This paper compares the stability of conformal-traceless and ADM formulations in 3D numerical relativity, demonstrating that the conformal-traceless approach offers improved long-term stability across various astrophysical scenarios.

Contribution

The authors develop and implement a conformal-traceless formulation that enhances the stability of 3D numerical evolutions of Einstein's equations in diverse gravitational systems.

Findings

Conformal-traceless approach yields more stable long-term evolutions.

The method performs well with weak and strong gravitational fields.

Short-term accuracy is slightly reduced compared to ADM.

Abstract

We study the stability of three-dimensional numerical evolutions of the Einstein equations, comparing the standard ADM formulation to variations on a family of formulations that separate out the conformal and traceless parts of the system. We develop an implementation of the conformal-traceless (CT) approach that has improved stability properties in evolving weak and strong gravitational fields, and for both vacuum and spacetimes with active coupling to matter sources. Cases studied include weak and strong gravitational wave packets, black holes, boson stars and neutron stars. We show under what conditions the CT approach gives better results in 3D numerical evolutions compared to the ADM formulation. In particular, we show that our implementation of the CT approach gives more long term stable evolutions than ADM in all the cases studied, but is less accurate in the short term for the range of resolutions used in our 3D simulations.