The dual twistor theory of self-dual black holes

TL;DR

This paper introduces a novel twistor-theoretic construction for self-dual black holes, revealing their geometric structures and deriving a new Kerr-Schild form of the Plebanski-Demianski metric.

Contribution

It presents a new approach using holomorphic quadrics in dual twistor space to describe self-dual black holes, clarifying their geometry and uncovering new metric forms.

Findings

New construction of self-dual black holes via dual twistor space

Explicit encoding of hyperkähler and Kerr-Schild structures

Discovery of a new Kerr-Schild form of Plebanski-Demianski metric

Abstract

The Taub-NUT and Eguchi-Hanson gravitational instantons, along with the self-dual Plebanski-Demianski metric, form a set of Euclidean metrics which can naturally be called `self-dual black holes', as they arise from self-dual slices of the most general vacuum, asymptotically flat black hole metric. These self-dual black holes are of interest for many reasons, and can famously be described through the non-linear graviton construction of twistor theory. However, the implicit nature of this twistor description obscures some features of the underlying geometry, particularly for the most general self-dual black holes. In this paper, we give a new construction of all asymptotically flat self-dual black holes based on holomorphic quadrics in flat dual twistor space, rather than the usual twistor space associated with self-duality. Remarkably, the geometry of the self-dual black holes -- including their hyperkahler structure, as well as Kerr-Schild and Gibbons-Hawking forms -- is directly encoded in the corresponding quadric. As a consequence, we obtain a previously unknown single Kerr-Schild form of the self-dual Plebanski-Demianski metric.