Gibbons–Hawking

Gibbons–Hawking
Name	Gibbons–Hawking
Field	Differential geometry, Theoretical physics, General relativity
Introduced by	Gary Gibbons, Stephen Hawking
Introduced date	1979
Notable examples	Taub–NUT metric, Eguchi–Hanson metric, ALE space

Gibbons–Hawking is a class of four-dimensional gravitational instanton solutions introduced by Gary Gibbons and Stephen Hawking that play a central role in studies of Euclidean General relativity, Quantum cosmology, and compactification in Kaluza–Klein theory. These solutions provide explicit metrics with self-dual or anti-self-dual Riemann curvature on noncompact and compact manifolds and have influenced work on Donaldson theory, Seiberg–Witten theory, and string-theoretic constructions such as Calabi–Yau manifold compactifications. The construction uses harmonic functions on flat three-dimensional space and yields multi-center geometries that interpolate between asymptotic behaviors like Asymptotically Locally Euclidean and Asymptotically Locally Flat.

Overview

The Gibbons–Hawking family generalizes the single-center Taub–NUT metric to multi-center configurations specified by positions in Euclidean R^3 and discrete charges. Each metric arises from a principal circle fibration over R^3 with connection determined by a harmonic function; this links the construction to classical harmonic analysis, the Laplace equation, and monopole moduli spaces studied by Michael Atiyah and Nigel Hitchin. In physical contexts these instantons contribute to path integrals in Euclidean quantum gravity and appear as gravitational analogues of Dirac monopole arrangements relevant to Yang–Mills theory and Supersymmetry.

Mathematical Formulation

Gibbons–Hawking metrics are hyper-Kähler four-manifolds obtained from a harmonic function V on R^3 with isolated singularities. Given V = ε + Σ_i (m_i / |x − x_i|) where ε ∈ {0,1} controls asymptotics and m_i are positive integers, one constructs a metric g = V^{-1} (dψ + A)^2 + V\, dx·dx where dA = * dV in R^3 and ψ parametrizes a circle fiber. This formulation uses results from Hodge theory, the theory of Self-dual manifolds, and the classification of gravitational instantons by Claude LeBrun and Simon Donaldson. Regularity at centers requires appropriate periodicity of ψ and links to orbifold singularities studied by John Milnor and resolutions in Algebraic geometry by Shigeru Mukai.

Physical Interpretations and Applications

In Kaluza–Klein theory the Gibbons–Hawking metrics model smooth compactifications with nontrivial circle fibrations, connecting to work by Theodor Kaluza and Oskar Klein and to modern M-theory compactifications studied by Edward Witten. They model gravitational effects of multi-monopole systems analogous to Bogomolny–Prasad–Sommerfield solutions and inform semiclassical evaluations in Euclidean path integral approaches used by Stephen Hawking himself and by researchers in Quantum field theory on curved spaces. In String theory and Mirror symmetry these metrics arise in local degenerations of Calabi–Yau threefolds and in constructions of D-brane moduli spaces investigated by Joseph Polchinski and Cumrun Vafa.

Examples and Notable Solutions

The single-center choice with V = 1 + m/|x| yields the Taub–NUT metric, related to the Gross–Perry–Sorkin monopole and used in studies by Gary Horowitz and Andrew Strominger. Taking ε = 0 with two centers produces the multi-Taub–NUT examples that, upon appropriate identifications, give rise to ALE space limits including the Eguchi–Hanson metric discovered by Toshihiro Eguchi and Andrew J. Hanson. Higher-multiplicity arrangements with symmetric placements of centers connect to discrete subgroups of SU(2) and the ADE classification appearing in works by Peter Kronheimer and Michael Atiyah on hyper-Kähler quotients. These explicit metrics serve as model geometries in analyses by S.-T. Yau and Eugenio Calabi related to existence theorems for Ricci-flat metrics.

Derivation and Construction Techniques

The derivation begins by imposing tri-holomorphic U(1) symmetry on a hyper-Kähler four-manifold and performing a Kaluza–Klein reduction to three dimensions; this approach uses methods from Hitchin's harmonic map framework and the Bogomolny equation for magnetic monopoles studied by Ernst Fischer and others. One solves the three-dimensional Laplace equation with point sources to obtain V, computes the connection 1-form A via the Hodge star in R^3, and reconstructs the four-metric ensuring self-duality of the curvature using identities from Spin geometry and the work of M. F. Atiyah on instantons. Regularity and global structure use patching considerations from Sheaf theory and orbifold resolution techniques by Miles Reid and William Fulton.

Relations to Other Gravitational Instantons

Gibbons–Hawking spaces relate closely to the Eguchi–Hanson metric, the Kronheimer construction of ALE gravitational instantons, and the broader classification of self-dual solutions by Nicolai Hitchin and C. N. Pope. Limits and degenerations connect to compact gravitational instantons examined by Claude LeBrun and to asymptotically locally flat examples like those used by G. W. Gibbons in black hole moduli studies alongside work on multicenter black hole solutions by Tom Banks and Andrew Strominger. The interplay with gauge-theoretic instantons studied in Donaldson theory and Seiberg–Witten theory highlights deep links between four-manifold topology, hyper-Kähler geometry, and quantum aspects of String theory.

Category:Gravitational instantons