Spectral approach to axisymmetric evolution of Einstein's equations

TL;DR

This paper introduces a spectral formulation for axisymmetric vacuum Einstein equations, utilizing spherical harmonics to handle coordinate singularities and proposing a new gauge compatible with spectral methods.

Contribution

The paper develops a spectral approach with a new gauge for axisymmetric Einstein equations, improving regularity handling and computational efficiency.

Findings

Spectral decomposition effectively manages axis singularities.

A new gauge compatible with spectral methods is proposed.

The formulation enhances computational stability and accuracy.

Abstract

We present a new formulation of Einstein's equations for an axisymmetric spacetime with vanishing twist in vacuum. We propose a fully constrained scheme and use spherical polar coordinates. A general problem for this choice is the occurrence of coordinate singularities on the axis of symmetry and at the origin. Spherical harmonics are manifestly regular on the axis and hence take care of that issue automatically. In addition a spectral approach has computational advantages when the equations are implemented. Therefore we spectrally decompose all the variables in the appropriate harmonics. A central point in the formulation is the gauge choice. One of our results is that the commonly used maximal-isothermal gauge turns out to be incompatible with tensor harmonic expansions, and we introduce a new gauge that is better suited. We also address the regularisation of the coordinate singularity at the origin.