Iterative Solution of the Kerr Black Hole Metric

TL;DR

This paper develops a recursive perturbative approach to derive the Kerr black hole metric as a double expansion in Newton's constant and spin parameter, addressing series resummation and regularization issues.

Contribution

It introduces a recursive solution method for the Kerr metric in harmonic gauge, enabling high-order expansions and analysis of series resummation challenges.

Findings

Recursive relations can be solved to arbitrary order in G and a.

Series resummation involves adding harmonic functions to coordinates.

Dimensional regularization of Fourier transforms is thoroughly analyzed.

Abstract

Using a recursive solution of the Einstein equations, we consider the perturbative expansion of the metric corresponding to a Kerr black hole. Because the metric is a function of two parameters, Newton's constant G and the Kerr spin parameter a, the perturbation theory naturally becomes a double expansion. In harmonic gauge the recursion relations can be solved to arbitrarily high orders in these two expansion parameters but to re-sum the series into the closed-form harmonic gauge metric requires the introduction of terms that are redundant and correspond to the addition of harmonic functions to the coordinates. Issues related to dimensional regularization of Fourier transforms are explained in detail.