Kaehler geometry of black holes and gravitational instantons

TL;DR

This paper derives a unified formula for the Kähler potential of various four-dimensional geometries, revealing relationships between black hole solutions, supergravity spacetimes, and complex structures through the Weyl double copy.

Contribution

It provides a closed-form expression for the Kähler potential of broad classes of geometries and uncovers new links between black hole solutions and complex geometric structures.

Findings

Kähler potentials for Schwarzschild and Kerr are related by a Newman-Janis shift

Supergravity black holes like Kerr-Sen are Hermitian but not conformal Kähler

Integrability conditions lead to the non-linear Weyl double copy

Abstract

We obtain a closed formula for the Kaehler potential of a broad class of four-dimensional Lorentzian or Euclidean conformal "Kaehler" geometries, including the Plebanski-Demianski class and various gravitational instantons such as Fubini-Study and Chen-Teo. We show that the Kaehler potentials of Schwarzschild and Kerr are related by a Newman-Janis shift. Our method also shows that a class of supergravity black holes, including the Kerr-Sen spacetime, is Hermitian (but not conformal Kaehler). We finally show that the integrability conditions of complex structures lead naturally to the (non-linear) Weyl double copy, and we give new vacuum and non-vacuum examples of this relation.