Kerr metric from two commuting complex structures

TL;DR

This paper simplifies the derivation of the Kerr metric using complex geometry, focusing on Euclidean metrics with commuting complex structures and Killing vectors, making the process more accessible and elementary.

Contribution

It provides a self-contained, elementary derivation of the Kerr metric based on complex structures and linear algebra, building on prior geometric methods.

Findings

Derivation of Kerr metric from complex geometric structures

Identification of a 2-parameter subfamily relevant to Kerr

Simplification of the construction process using linear algebra

Abstract

The main aim of this paper is to simplify and popularise the construction from the 2013 paper by Apostolov, Calderbank, and Gauduchon, which (among other things) derives the Plebanski-Demianski family of solutions of GR using ideas of complex geometry. The starting point of this construction is the observation that the Euclidean versions of these metrics should have two different commuting complex structures, as well as two commuting Killing vector fields. After some linear algebra, this leads to an ansatz for the metrics, which is half-way to their complete determination. Kerr metric is a special 2-parameter subfamily in this class, which makes these considerations directly relevant to Kerr as well. This results in a derivation of the Kerr metric that is self-contained and elementary, in the sense of being mostly an exercise in linear algebra.