On the Numerical Integration of Einstein's Field Equations

TL;DR

The paper compares the standard ADM (System I) and a conformal, connection-function–based reformulation (System II) of Einstein's equations in 3+1 form. By applying a York–Lichnerowicz split and introducing conformal variables $\tilde{\gamma}_{ij}$, $\tilde{A}_{ij}$, $\phi$, and the conformal connection functions $\tilde{\Gamma}^i$, the Ricci operator becomes elliptic and the evolution reduces to coupled wave equations, yielding markedly improved numerical stability for small-amplitude gravitational waves across geodesic and harmonic slicings. The authors implement both systems identically, use Crank–Nicolson time stepping with Sommerfeld boundaries, and demonstrate that System II can evolve for hundreds of light-crossing times, while System I crashes early, highlighting the profound impact of mathematical formulation on numerical behavior. The results advocate adopting conformal-based 3+1 formulations in 3D numerical relativity, potentially simplifying stability concerns and guiding boundary-condition treatments in future simulations with matter or strong fields.

Abstract

Many numerical codes now under development to solve Einstein's equations of general relativity in 3+1 dimensional spacetimes employ the standard ADM form of the field equations. This form involves evolution equations for the raw spatial metric and extrinsic curvature tensors. Following Shibata and Nakamura, we modify these equations by factoring out the conformal factor and introducing three ``connection functions''. The evolution equations can then be reduced to wave equations for the conformal metric components, which are coupled to evolution equations for the connection functions. We evolve small amplitude gravitational waves and make a direct comparison of the numerical performance of the modified equations with the standard ADM equations. We find that the modified form exhibits much improved stability.

Figures (3)

• Figure 1: Evolution of the trace of the extrinsic curvature $K$ for a small amplitude wave in geodesic slicing at the origin (see text for details). The solid line is the result for System II, and the dashed line for System I. The dotted line is the approximate solution (\ref{['K_approx']}).
• Figure 2: Evolution of the extrinsic curvature component $K_{zz}$ at the origin in geodesic slicing. The solid line is the result for System II, and the dashed line for System I. For System II, we constructed $K_{zz}$ from $\tilde{A}_{zz}$, $\phi$, $K$ and $\tilde{\gamma}_{zz}$.
• Figure 3: Evolution of the extrinsic curvature component $K_{zz}$ at the origin in harmonic slicing. The solid line is the result for System II, and the dashed line for System I. For System II, we constructed $K_{zz}$ from $\tilde{A}_{zz}$, $\phi$, $K$ and $\tilde{\gamma}_{zz}$.