Slowly rotating Kerr metric derived from the Einstein equations in affine-null coordinates

TL;DR

This paper derives a slowly rotating Kerr metric from Einstein's equations using affine-null coordinates, providing high-order approximations and verifying equivalence with the Kerr metric in the slow rotation limit.

Contribution

It introduces a new method to approximate the Kerr metric in affine-null coordinates through high-order perturbations and confirms its consistency with the classical Kerr solution.

Findings

Derived high-order metric approximations for slowly rotating Kerr spacetime.

Established the equivalence between the affine-null coordinate metric and Kerr metric.

Validated the approach through explicit coordinate transformation.

Abstract

Using a quasi-spherical approximation of an affine-null metric adapted to an asymptotic Bondi inertial frame, we present high order approximations of the metric functions in terms of the specific angular momentum for a slowly rotating stationary and axi-symmetric vacuum spacetime. The metric is obtained by following the procedure of integrating the hierarchy of Einstein equations in a characteristic formulation utilizing master functions for the perturbations. It is further verified its equivalence with the Kerr metric in the slowly rotation approximation by carrying out an explicit transformation between the Boyer-Lindquist coordinates to the employed affine-null coordinates.