Notes on causal differencing in ADM/CADM formulations: a 1D comparison

TL;DR

This paper compares causal differencing in ADM and conformal ADM (CADM) formulations for simulating black hole spacetimes, focusing on their effectiveness in 1D models and potential for excision techniques.

Contribution

It introduces a formulation of causal differencing compatible with CADM and compares its performance with ADM in one-dimensional black hole simulations.

Findings

CADM shows longer evolution times than ADM in 1D models.

CADM has better boundary dependence in simulations.

The paper provides a framework for applying causal differencing to CADM.

Abstract

Causal differencing has shown to be one of the promising and successful approaches towards excising curvature singularities from numerical simulations of black hole spacetimes. So far it has only been actively implemented in the ADM and Einstein-Bianchi 3+1 formulations of the Einstein equations. Recently, an approach closely related to the ADM one, commonly referred to as as ``conformal ADM'' (CADM) has shown excellent results when modeling waves on flat spacetimes and black hole spacetimes where singularity avoiding slices are used to deal with the singularity. In these cases, the use of CADM has yielded longer evolutions and better outer boundary dependence than those obtained with the ADM one. If this success translates to the case where excision is implemented, then the CADM formulation will likely be a prime candidate for modeling generic black hole spacetimes. In the present work we investigate the applicability of causal differencing to CADM, presenting the equations in a convenient way for such a goal and compare its application with the ADM approach in 1D.