Generalized Harmonic Equations in 3+1 Form

TL;DR

This paper reformulates the generalized harmonic equations of general relativity in 3+1 form, enabling direct comparison with ADM and BSSN formulations, and explores their structure in conformal variables.

Contribution

It presents a new 3+1 formulation of the generalized harmonic equations, facilitating comparisons with ADM and BSSN approaches in numerical relativity.

Findings

Derived a 3+1 PDE system for harmonic equations

Compared harmonic, ADM, and BSSN formulations

Expressed equations in conformal variables

Abstract

The generalized harmonic equations of general relativity are written in 3+1 form. The result is a system of partial differential equations with first order time and second order space derivatives for the spatial metric, extrinsic curvature, lapse function and shift vector, plus fields that represent the time derivatives of the lapse and shift. This allows for a direct comparison between the generalized harmonic and the Arnowitt-Deser-Misner formulations. The 3+1 generalized harmonic equations are also written in terms of conformal variables and compared to the Baumgarte-Shapiro-Shibata-Nakamura equations with moving puncture gauge conditions.