On the Numerical Integration of Einstein's Field Equations

TL;DR

This paper introduces a modified formulation of Einstein's field equations for numerical relativity that improves stability by evolving conformal metric components and connection functions, demonstrated through gravitational wave simulations.

Contribution

The paper presents a new formulation of Einstein's equations that enhances numerical stability by factoring out the conformal factor and using connection functions, compared to the standard ADM form.

Findings

Modified equations show much improved stability.

Wave equations for conformal metric components are effective.

Comparison with standard ADM demonstrates benefits.

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.