Gibbons–Hawking ansatz

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

Description

Suppose that is an open subset of, and let denote the Hodge star operator on with respect to the usual 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.

Examples

Quaternions

The usual 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.

ALE gravitational instantons

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.