Numerical validation of the Kerr metric in Bondi-Sachs form

TL;DR

The paper presents a method to obtain the Kerr metric in Bondi-Sachs form by a coordinate transformation from Pretorius-Israel coordinates and validates the resulting metric by numerically computing the Ricci tensor on a regular grid, confirming Ricci-flatness and axis regularity. It derives explicit relations between the PI and Bondi-Sachs coordinates, and constructs the Bondi-Sachs Kerr metric with a carefully chosen function $H(r_*,\theta_*)$ to ensure the correct form. The authors perform detailed numerical experiments, achieving second-order convergence of the Ricci tensor to zero and confirming the PI metric’s Ricci-flatness as well, while showing a competing Fletcher-Lun representation is not regular at the pole. The work highlights practical steps and remaining challenges for using Bondi-Sachs Kerr in numerical relativity, including the need for an area-coordinate radial variable and stereographic angular coordinates to enable robust pole behavior. Overall, it provides a concrete, validated pathway toward a numerically stable Bondi-Sachs Kerr representation and benchmarks several candidate metrics.

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.