Kerr metric from two commuting complex structures

Krasnov, Kirill; Shaw, Adam

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 Pleba´nskiPleba´nski-Demia´nskiDemia´nski 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 halfway 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.

Citation

Krasnov, K., & Shaw, A. (2025). Kerr metric from two commuting complex structures. Classical and Quantum Gravity, 42(6), Article 065013. https://doi.org/10.1088/1361-6382/adb531

• Journal Article Type: Article
• Acceptance Date: Feb 12, 2025
• Online Publication Date: Mar 3, 2025
• Publication Date: Mar 21, 2025
• Deposit Date: Feb 14, 2025
• Publicly Available Date: Feb 14, 2025
• Journal: Classical and Quantum Gravity
• Print ISSN: 0264-9381
• Electronic ISSN: 1361-6382
• Publisher: IOP Publishing
• Peer Reviewed: Peer Reviewed
• Volume: 42
• Issue: 6
• Article Number: 065013
• DOI: https://doi.org/10.1088/1361-6382/adb531
• Public URL: https://nottingham-repository.worktribe.com/output/45313742
• Publisher URL: https://iopscience.iop.org/article/10.1088/1361-6382/adb531