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Gravitational Instantons, old and new

Maciej Dunajski

TL;DR

This review surveys gravitational instantons—complete 4D Riemannian solutions to Einstein (and Einstein–Maxwell) equations with finite action and flat-like asymptotics—through concrete examples and unifying geometric frameworks. It develops the hyper–Kähler and Gibbons–Hawking perspectives for multi-centered constructions, and highlights the Chen–Teo instanton as a key explicit toric family with a detailed rod structure, Yang–equation reduction, and twistorial interpretation. The article also surveys broader developments, including the ALE/ALF/ALG/ALH taxonomy, Einstein–Maxwell instantons, and twistor theory, illustrating how these tools enrich both mathematics and Euclidean quantum gravity. Overall, the work clarifies how explicit metrics, rod data, and twistorial methods jointly illuminate the landscape of gravitational instantons and their physical and geometric significance.

Abstract

This is a review of gravitational instantons -- solutions to Riemannian Einstein or Einstein-Maxwell equations in four dimensions which yield complete metrics on non-compact four-manifolds, and which asymptotically `look like' flat space. The review focuses on examples, and is based on lectures given by the author at the Cracow School of Theoretical Physics held in Zakopane in June 2024.

Gravitational Instantons, old and new

TL;DR

This review surveys gravitational instantons—complete 4D Riemannian solutions to Einstein (and Einstein–Maxwell) equations with finite action and flat-like asymptotics—through concrete examples and unifying geometric frameworks. It develops the hyper–Kähler and Gibbons–Hawking perspectives for multi-centered constructions, and highlights the Chen–Teo instanton as a key explicit toric family with a detailed rod structure, Yang–equation reduction, and twistorial interpretation. The article also surveys broader developments, including the ALE/ALF/ALG/ALH taxonomy, Einstein–Maxwell instantons, and twistor theory, illustrating how these tools enrich both mathematics and Euclidean quantum gravity. Overall, the work clarifies how explicit metrics, rod data, and twistorial methods jointly illuminate the landscape of gravitational instantons and their physical and geometric significance.

Abstract

This is a review of gravitational instantons -- solutions to Riemannian Einstein or Einstein-Maxwell equations in four dimensions which yield complete metrics on non-compact four-manifolds, and which asymptotically `look like' flat space. The review focuses on examples, and is based on lectures given by the author at the Cracow School of Theoretical Physics held in Zakopane in June 2024.
Paper Structure (19 sections, 3 theorems, 50 equations)

This paper contains 19 sections, 3 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Examples
  3. Euclidean Schwarzschild metric
  4. Anti--self--dual Taub--NUT and ALF metric
  5. Eguchi--Hanson and the ALE metrics
  6. Multi--centered metrics
  7. Mathematical detour: Hyper--Kähler metrics
  8. Gibbons--Hawking ansatz
  9. The Chen--Teo instanton
  10. Explicit formulae
  11. The rod structure
  12. The Yang equation and ASDYM
  13. Twistor construction
  14. Twistor bundle for toric Ricci flat metrics
  15. Other developments
  16. ...and 4 more sections

Key Result

Theorem 2.3

For any $\Gamma$ (cyclic $A_N$, dihedral $D_N$, dihedral, tetrahedral, octahedral, and icosahedral) there exists an ALE gravitational instanton.

Theorems & Definitions (6)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Kronheimer K1K2
  • Definition 3.1
  • Theorem 4.1: Ward W1
  • Theorem 5.1: Pe76, AHS78