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

Abstract

Using a quasispherical 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 axisymmetric 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 further verified its equivalence with the Kerr metric in the slowly rotating approximation by carrying out an explicit transformation between the Boyer-Lindquist coordinates to the employed affine-null coordinates. A peculiar feature of the derivation is that in the solution of the perturbation equations for every order a new integration constant appears which cannot be set to zero using asymptotical flatness or regularity arguments. However, these additional integration constants can be absorbed into the overall Komar mass and Komar angular momentum of a slowly rotating black hole.