A Simple, Direct Finite Differencing of the Einstein Equations

TL;DR

This paper presents a straightforward variation of the Generalized Harmonic method that simplifies Einstein equation evolution, enabling accurate simulations of compact objects and facilitating the integration of relativistic effects into astrophysical models.

Contribution

It introduces a simple splitting technique separating wave equations from the Ricci tensor, improving simulation accuracy for strong gravitational fields.

Findings

Accurate simulation of neutron star-level gravitational fields.

Simplified method for incorporating relativistic effects into astrophysics.

Potential application to core collapse and accretion disk modeling.

Abstract

We investigate a simple variation of the Generalized Harmonic method for evolving the Einstein equations. A flat space wave equation for metric perturbations is separated from the Ricci tensor, with the rest of the Ricci tensor becoming a source for these wave equations. We demonstrate that this splitting method allows for the accurate simulation of compact objects, with gravitational field strengths less than or equal to those of neutron stars. This method could thus provide a straightforward path for general relativistic effects to be added to astrophysics simulations, such as in core collapse, accretion disks, and extreme mass ratio systems.