Numerical validation of the Kerr metric in Bondi-Sachs form

TL;DR

This paper numerically validates the Kerr metric in Bondi-Sachs form by transforming, discretizing, and analyzing its Ricci tensor, confirming regularity and Ricci-flatness, and comparing different representations.

Contribution

It provides a numerical validation of the Kerr metric in Bondi-Sachs coordinates, confirming regularity and Ricci-flatness, and evaluates different metric representations.

Findings

Ricci tensor converges to zero across discretizations

Confirmed regularity near the axis of symmetry

Identified irregular behavior at the pole in a previous representation

Abstract

A metric representing the Kerr geometry has been obtained by Pretorius and Israel. We make a coordinate transformation on this metric, thereby bringing it into Bondi-Sachs form. In order to validate the metric, we evaluate it numerically on a regular grid of the new coordinates. The Ricci tensor is then computed, for different discretizations, and found to be convergent to zero. We also investigate the behaviour of the metric near the axis of symmetry and confirm regularity. Finally we investigate a Bondi-Sachs representation of the Kerr geometry reported by Fletcher and Lun; we confirm numerically that their metric is Ricci flat, but find that it has an irregular behaviour at the pole.