Harmonic coordinate method for simulating generic singularities

TL;DR

This paper introduces a numerical harmonic coordinate method for simulating Einstein's equations with scalar matter, revealing that near singularities, the dynamics resemble Kasner spacetime behavior.

Contribution

It develops a harmonic coordinate-based numerical approach for Einstein's equations with scalar fields and applies it to study singularity behavior.

Findings

Near singularities, dynamics are locally Kasner-like.

Harmonic coordinates simplify Einstein's equations to wave-like form.

The method enables detailed exploration of generic singularity approaches.

Abstract

This paper presents both a numerical method for general relativity and an application of that method. The method involves the use of harmonic coordinates in a 3+1 code to evolve the Einstein equations with scalar field matter. In such coordinates, the terms in Einstein's equations with the highest number of derivatives take a form similar to that of the wave equation. The application is an exploration of the generic approach to the singularity for this type of matter. The preliminary results indicate that the dynamics as one approaches the singularity is locally the dynamics of the Kasner spacetimes.