Numerical integration of Einstein’s field equations

TLDR

Numerical relativity codes commonly use the ADM formulation, which evolves the spatial metric and extrinsic curvature in 3+1 dimensions. The study aims to modify the ADM equations by factoring out the conformal factor and adding three connection functions. The modified system reduces to wave equations for the conformal metric coupled to connection functions, and is tested by evolving small‑amplitude gravitational waves to compare its performance with standard ADM. The modified equations show significantly improved numerical stability compared to standard ADM.

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.