Gibbons–Hawking ansatz

Short description: Method in general relativity

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 $U$ is an open subset of ${\mathbb{ℝ}}^3$, and let $*$ denote the Hodge star operator on ${\mathbb{ℝ}}^3$ with respect to the usual (flat) Euclidean metric. $V$ is a harmonic function defined on $U$ such that the cohomology class $[\frac{1}{2\pi }*dV]$ is integral, i.e. lies in the image of $H^2(U;\mathbb{\mathbb{Z}})\hookrightarrow H^2(U;\mathbb{ℝ})$. Then there is a $U(1)$-principal bundle $\pi :P\to U$ equipped with a connection 1-form $\eta \in {\mathrm{\Omega }}^1(P;\mathfrak{𝔲}(1))$ whose curvature form is $d\eta ={\pi }^{*}(*dV)$. Then the Riemannian metric $$g=V\sum _{j=1}^{3}d{x}_j\otimes d{x}_j+\frac{1}{V}\eta \otimes \eta$$ is hyperkahler, and typically extends to the boundary of $U$.

Examples

Quaternions

The usual (flat) metric on the quaternions $\mathbb{ℍ}\cong {\mathbb{ℂ}}^2$ is hyperkahler. It can be obtained as a result of the Gibbons-Hawking ansatz applied to the open subset $U={\mathbb{ℝ}}^3\setminus \{0\}$ and the harmonic function $V(x)=\frac{1}{2|x|}$.

ALE gravitational instantons

The ALE gravitational instanton of type $A_{k-1}$ can be obtained by applying the Gibbons-Hawking ansatz to the open subset $U={\mathbb{ℝ}}^3\setminus \{p_1,\dots ,p_k\}$ for $k$ distinct collinear points $p_1,\dots ,p_k$ and the harmonic function $V(x)=\sum _{j=1}^k\frac{1}{2|x-p_j|}$. In the case $k=2$, we recover the Eguchi-Hanson metric on $T^{*}{\mathbb{\mathbb{P}}}^1$.

See also

• Gibbons–Hawking space
• Ooguri–Vafa metric

References

• Gibbons, G.W.; Hawking, S. W. (1978), "Gravitational multi-instantons", Physics Letters B 78 (4): 430–432, doi:10.1016/0370-2693(78)90478-1, ISSN 0370-2693, Bibcode: 1978PhLB...78..430G 
• Gibbons, G. W.; Hawking, S. W. (1979), "Classification of gravitational instanton symmetries", Communications in Mathematical Physics 66 (3): 291–310, doi:10.1007/bf01197189, ISSN 0010-3616, Bibcode: 1979CMaPh..66..291G, http://projecteuclid.org/getRecord?id=euclid.cmp/1103905051 
• Gonzalo Pérez, Jesús; Geiges, Hansjörg (2010), "A homogeneous Gibbons–Hawking ansatz and Blaschke products", Advances in Mathematics 225 (5): 2598–2615, doi:10.1016/j.aim.2010.05.006, ISSN 0001-8708