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

TL;DR

This work addresses the instability of 3D numerical relativity evolutions under the standard ADM formulation by adopting a conformal-traceless (CT) decomposition that evolves a conformal metric, the trace-free extrinsic curvature, and related auxiliary variables. By systematically testing CT implementations (notably AF2 for maximal slicing and AFA for algebraic slicings) against ADM across vacuum and matter spacetimes—including Brill waves, Misner data, boson stars, and static neutron stars—the authors demonstrate enhanced long-term stability with CT, albeit with some reduction in short-term accuracy at fixed resolution. The study also exposes the trade-offs between stability and precision and discusses boundary conditions, gauge choices, and the role of actively enforcing constraints during evolution. Overall, the CT approach improves robustness for strong-field 3D evolutions and points to productive directions for combining CT with hyperbolic formulations and refined gauge strategies to extend reliable simulations of compact-object dynamics and gravitational-wave generation.

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.

Figures (15)

• Figure 1: a) Evolution of the minimum value of the lapse for an axisymmetric Brill wave data set with $a$=4, using the standard ADM formulation with maximal slicing. The simulation crashes at $t$=8 with a catastrophic collapse of the lapse. b) Evolution of the maximum value of the trace of the extrinsic curvature $K$.
• Figure 2: a) Evolution of the minimum value of the lapse for an axisymmetric Brill wave data set with $a$=4, using the standard ADM formulation with maximal slicing and a K-driver. The simulation goes somewhat further, and now crashes at $t$=9 with a catastrophic blow-up of the lapse. b) Evolution of the maximum value of the trace of the extrinsic curvature $K$. The trace now remains much smaller during the simulation.
• Figure 3: Evolution of the minimum value of the lapse for an axisymmetric Brill wave data set with $a$=4, using the 6 different variations of the CT system described in the text.
• Figure 4: Evolution of the minimum value of the lapse for an axisymmetric Brill wave data set with $a$=0.01 for the ADM system and three variations of the CT system (Mom, AFK, AFA). Notice that since the lapse remains very close to 1, we are in fact plotting $(\alpha-1)\times10^5$.
• Figure 5: L2-norms of the hamiltonian constraint for the $a$=0.01 and $a$=4 cases, using the ADM (solid line) and the AF2 systems (dashed line).