Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Kerr metric from two commuting complex structures

Kirill Krasnov, Adam Shaw

TL;DR

The paper develops an elementary, geometry-driven derivation of the Euclidean Kerr metric by exploiting ambitoric (double-Kähler) geometry arising from two commuting complex structures and two commuting Killing fields, following and simplifying Apostolov–Calderbank–Gauduchon’s framework. Using the Plebanski chiral formalism, it shows how one-sided type D metrics are conformal to Kähler and extends this to double-sided type D spacetimes with toric symmetry. The derivation yields the Plebański-Demiański family and specializes to Kerr via simple parameter choices, all within an explicit linear-algebraic metric ansatz. This framework clarifies Kerr’s integrability structure and holds promise for gravitational perturbations and alternative connection-based formulations of 4D GR.

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.

Kerr metric from two commuting complex structures

TL;DR

The paper develops an elementary, geometry-driven derivation of the Euclidean Kerr metric by exploiting ambitoric (double-Kähler) geometry arising from two commuting complex structures and two commuting Killing fields, following and simplifying Apostolov–Calderbank–Gauduchon’s framework. Using the Plebanski chiral formalism, it shows how one-sided type D metrics are conformal to Kähler and extends this to double-sided type D spacetimes with toric symmetry. The derivation yields the Plebański-Demiański family and specializes to Kerr via simple parameter choices, all within an explicit linear-algebraic metric ansatz. This framework clarifies Kerr’s integrability structure and holds promise for gravitational perturbations and alternative connection-based formulations of 4D GR.

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.
Paper Structure (29 sections, 10 theorems, 148 equations)

This paper contains 29 sections, 10 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Chiral Formalism
  4. Plebanski action and field equations
  5. SU(2) Structures and Killing Vectors
  6. One-sided type D metrics are conformal to Kähler
  7. Derdziński theorem
  8. Killing Vectors in Einstein one-sided type D spaces
  9. Two complex structures of Euclidean Kerr
  10. Metric and Chiral objects
  11. Self-dual objects
  12. Anti-self-dual objects
  13. Conformal to Kähler
  14. Complex Structures
  15. Killing Vectors
  16. ...and 14 more sections

Key Result

Lemma 1

$\Sigma$-Killing vectors $X \in TM$ can be equivalently described as those satisfying where $d^A$ is the exterior covariant derivative with respect to the $\Sigma$-compatible $d^A \Sigma^i=0$ connection $A^i$.

Theorems & Definitions (20)

  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Theorem 1
  • Proposition 2
  • proof : Proof of \ref{['rem:Einstein_Divergence_Free']}
  • proof : Proof of \ref{['thrm:DerdziConfKahler']}
  • Proposition 3
  • proof
  • ...and 10 more