In mathematics, the Gibbons–Hawking ansatz is a method of constructing gravitational instantons introduced by Gary Gibbons and Stephen Hawking (1978, 1979). It gives examples of hyperkähler manifolds in dimension 4 that are invariant under a circle action.

Suppose that is an open subset of , and let denote the Hodge star operator on with respect to the usual (flat) Euclidean metric. is a harmonic function defined on such that the cohomology class is integral, i.e. lies in the image of . Then there is a -principal bundle equipped with a connection 1-form whose curvature form is . Then the Riemannian metric is hyperkahler, and typically extends to the boundary of .

The usual (flat) metric on the quaternions is hyperkahler. It can be obtained as a result of the Gibbons-Hawking ansatz applied to the open subset and the harmonic function .

The ALE gravitational instanton of type can be obtained by applying the Gibbons-Hawking ansatz to the open subset for distinct collinear points and the harmonic function . In the case , we recover the Eguchi-Hanson metric on .

• Gibbons–Hawking space
• Ooguri–Vafa metric

• Gibbons, G.W.; Hawking, S. W. (1978), "Gravitational multi-instantons", Physics Letters B, 78 (4): 430–432, Bibcode:1978PhLB...78..430G, doi:10.1016/0370-2693(78)90478-1, ISSN 0370-2693
• Gibbons, G. W.; Hawking, S. W. (1979), "Classification of gravitational instanton symmetries", Communications in Mathematical Physics, 66 (3): 291–310, Bibcode:1979CMaPh..66..291G, doi:10.1007/bf01197189, ISSN 0010-3616, MR 0535152, S2CID 123183399
• Gonzalo Pérez, Jesús; Geiges, Hansjörg (2010), "A homogeneous Gibbons–Hawking ansatz and Blaschke products", Advances in Mathematics, 225 (5): 2598–2615, arXiv:0807.0086, doi:10.1016/j.aim.2010.05.006, ISSN 0001-8708, MR 2680177