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.
Description Suppose that U {\displaystyle U} is an open subset of R 3 {\displaystyle \mathbb {R} ^{3}} , and let ∗ {\displaystyle *} denote the Hodge star operator on R 3 {\displaystyle \mathbb {R} ^{3}} with respect to the usual (flat) Euclidean metric. V {\displaystyle V} is a harmonic function defined on U {\displaystyle U} such that the cohomology class [ 1 2 π ∗ d V ] {\displaystyle \left[{\frac {1}{2\pi }}*dV\right]} is integral, i.e. lies in the image of H 2 ( U ; Z ) ↪ H 2 ( U ; R ) {\displaystyle H^{2}(U;\mathbb {Z} )\hookrightarrow H^{2}(U;\mathbb {R} )} . Then there is a U ( 1 ) {\displaystyle U(1)} -principal bundle π : P → U {\displaystyle \pi :P\to U} equipped with a connection 1-form η ∈ Ω 1 ( P ; u ( 1 ) ) {\displaystyle \eta \in \Omega ^{1}(P;{\mathfrak {u}}(1))} whose curvature form is d η = π ∗ ( ∗ d V ) {\displaystyle d\eta =\pi ^{*}(*dV)} . Then the Riemannian metric
g = V ∑ j = 1 3 d x j ⊗ d x j + 1 V η ⊗ η {\displaystyle g=V\sum _{j=1}^{3}dx_{j}\otimes dx_{j}+{\frac {1}{V}}\eta \otimes \eta }
is hyperkahler, and typically extends to the boundary of U {\displaystyle U} .
Examples
Quaternions The usual (flat) metric on the quaternions H ≅ C 2 {\displaystyle \mathbb {H} \cong \mathbb {C} ^{2}} is hyperkahler. It can be obtained as a result of the Gibbons-Hawking ansatz applied to the open subset U = R 3 ∖ { 0 } {\displaystyle U=\mathbb {R} ^{3}\setminus \{0\}} and the harmonic function V ( x ) = 1 2 | x | {\displaystyle V(x)={\frac {1}{2|x|}}} .
ALE gravitational instantons The ALE gravitational instanton of type A k − 1 {\displaystyle A_{k-1}} can be obtained by applying the Gibbons-Hawking ansatz to the open subset U = R 3 ∖ { p 1 , … , p k } {\displaystyle U=\mathbb {R} ^{3}\setminus \{p_{1},\ldots ,p_{k}\}} for k {\displaystyle k} distinct collinear points p 1 , … , p k {\displaystyle p_{1},\ldots ,p_{k}} and the harmonic function V ( x ) = ∑ j = 1 k 1 2 | x − p j | {\displaystyle V(x)=\sum _{j=1}^{k}{\frac {1}{2|x-p_{j}|}}} . In the case k = 2 {\displaystyle k=2} , we recover the Eguchi-Hanson metric on T ∗ P 1 {\displaystyle T^{*}\mathbb {P} ^{1}} .
See also Gibbons–Hawking space Ooguri–Vafa metric
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