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The dual twistor theory of self-dual black holes

Tim Adamo, Bernardo Araneda, Sean Seet

TL;DR

This work develops a complete, explicit map between holomorphic quadrics in flat dual twistor space and self-dual black hole geometries, showing that every such metric is encoded in flat-space data. It leverages the Penrose transform and Tod's Kerr-Schild construction to produce SD hyperkähler metrics that admit both Kerr-Schild and Gibbons-Hawking forms, with the dual twistor quadric determining all Killing spinors and symmetries. The authors classify generic dual twistor quadrics into three cases, revealing a precise one-to-one correspondence with self-dual Taub-NUT, Eguchi-Hanson, and self-dual Plebański-Demiański metrics, and they demonstrate that each case yields a single Kerr-Schild representation. This framework unifies previously known SDBHs under a flat-twistor data paradigm and provides a concrete platform for exploring perturbations and holographic connections in self-dual backgrounds. The results have potential implications for analytic gravitational scattering on self-dual backgrounds and for linking twistor methods with Teukolsky-type perturbation analyses in black hole physics.

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.

The dual twistor theory of self-dual black holes

TL;DR

This work develops a complete, explicit map between holomorphic quadrics in flat dual twistor space and self-dual black hole geometries, showing that every such metric is encoded in flat-space data. It leverages the Penrose transform and Tod's Kerr-Schild construction to produce SD hyperkähler metrics that admit both Kerr-Schild and Gibbons-Hawking forms, with the dual twistor quadric determining all Killing spinors and symmetries. The authors classify generic dual twistor quadrics into three cases, revealing a precise one-to-one correspondence with self-dual Taub-NUT, Eguchi-Hanson, and self-dual Plebański-Demiański metrics, and they demonstrate that each case yields a single Kerr-Schild representation. This framework unifies previously known SDBHs under a flat-twistor data paradigm and provides a concrete platform for exploring perturbations and holographic connections in self-dual backgrounds. The results have potential implications for analytic gravitational scattering on self-dual backgrounds and for linking twistor methods with Teukolsky-type perturbation analyses in black hole physics.

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.
Paper Structure (19 sections, 10 theorems, 168 equations)

This paper contains 19 sections, 10 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Self-dual black hole metrics
  3. Review: the Plebański-Demiański metrics
  4. Self-dual Plebański-Demiański
  5. Degenerate case:
  6. Eguchi-Hanson
  7. Self-dual Taub-NUT
  8. From dual twistor quadrics to SD Kerr-Schild metrics
  9. From dual twistor quadrics to SD Kerr-Schild metrics
  10. Twistor and dual twistor spaces
  11. From self-dual null Maxwell fields to self-dual Kerr-Schild metrics
  12. Self-dual null Maxwell fields from dual twistor quadrics
  13. Gibbons-Hawking metrics from dual twistor quadrics
  14. Dual twistor quadrics are self-dual black holes
  15. Classifying dual twistor quadrics
  16. ...and 4 more sections

Key Result

Theorem 1

There is a one-to-one correspondence between generic holomorphic quadrics in the dual twistor space of flat space, compatible with Euclidean reality conditions, and self-dual black hole metrics.

Theorems & Definitions (12)

  • Theorem 1
  • Definition 2.1: Self-dual black hole
  • Theorem 3.1: Tod Tod:1982mmp
  • Definition 3.1: Dual twistor quadric
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 4.1
  • Corollary 4.1
  • Proposition 4.2
  • ...and 2 more