Taub–NUT metric

${\displaystyle ds^{2}={\frac {1}{4}}{\frac {r+n}{r-n}}dr^{2}+{\frac {r-n}{r+n}}n^{2}{\sigma _{3}}^{2}+{\frac {1}{4}}(r^{2}-n^{2})({\sigma _{1}}^{2}+{\sigma _{2}}^{2})}$

Eguchi–Hanson metric

The Eguchi–Hanson space is defined by a metric the cotangent bundle of the 2-sphere ${\displaystyle T^{*}\mathbb {CP} (1)=T^{*}S^{2}}$. This metric is

${\displaystyle ds^{2}=\left(1-{\frac {a}{r^{4}}}\right)^{-1}dr^{2}+{\frac {r^{2}}{4}}\left(1-{\frac {a}{r^{4}}}\right){\sigma _{3}}^{2}+{\frac {r^{2}}{4}}(\sigma _{1}^{2}+\sigma _{2}^{2}).}$

where ${\displaystyle r\geq a^{1/4}}$. This metric is smooth everywhere if it has no conical singularity at ${\displaystyle r\rightarrow a^{1/4}}$, ${\displaystyle \theta =0,\pi }$. For ${\displaystyle a=0}$ this happens if ${\displaystyle \psi }$ has a period of ${\displaystyle 4\pi }$, which gives a flat metric on R⁴; However, for ${\displaystyle a\neq 0}$ this happens if ${\displaystyle \psi }$ has a period of ${\displaystyle 2\pi }$.

Asymptotically (i.e., in the limit ${\displaystyle r\rightarrow \infty }$) the metric looks like

${\displaystyle ds^{2}=dr^{2}+{\frac {r^{2}}{4}}\sigma _{3}^{2}+{\frac {r^{2}}{4}}(\sigma _{1}^{2}+\sigma _{2}^{2})}$

which naively seems as the flat metric on R⁴. However, for ${\displaystyle a\neq 0}$, ${\displaystyle \psi }$ has only half the usual periodicity, as we have seen. Thus the metric is asymptotically R⁴ with the identification ${\displaystyle \psi \,{\sim }\,\psi +2\pi }$, which is a Z₂ subgroup of SO(4), the rotation group of R⁴. Therefore, the metric is said to be asymptotically R⁴/Z₂.

There is a transformation to another coordinate system, in which the metric looks like

${\displaystyle ds^{2}={\frac {1}{V(\mathbf {x} )}}(d\psi +{\boldsymbol {\omega }}\cdot d\mathbf {x} )^{2}+V(\mathbf {x} )d\mathbf {x} \cdot d\mathbf {x} ,}$

where ${\displaystyle \nabla V=\pm \nabla \times {\boldsymbol {\omega }},\quad V=\sum _{i=1}^{2}{\frac {1}{|\mathbf {x} -\mathbf {x} _{i}|}}.}$

(For a = 0, ${\displaystyle V={\frac {1}{|\mathbf {x} |}}}$, and the new coordinates are defined as follows: one first defines ${\displaystyle \rho =r^{2}/4}$ and then parametrizes ${\displaystyle \rho }$, ${\displaystyle \theta }$ and ${\displaystyle \phi }$ by the R³ coordinates ${\displaystyle \mathbf {x} }$, i.e. ${\displaystyle \mathbf {x} =(\rho \sin \theta \cos \phi ,\rho \sin \theta \sin \phi ,\rho \cos \theta )}$).

In the new coordinates, ${\displaystyle \psi }$ has the usual periodicity ${\displaystyle \psi \ {\sim }\ \psi +4\pi .}$

One may replace V by

${\displaystyle \quad V=\sum _{i=1}^{n}{\frac {1}{|\mathbf {x} -\mathbf {x} _{i}|}}.}$

For some n points ${\displaystyle \mathbf {x} _{i}}$, i = 1, 2..., n. This gives a multi-center Eguchi–Hanson gravitational instanton, which is again smooth everywhere if the angular coordinates have the usual periodicities (to avoid conical singularities). The asymptotic limit (${\displaystyle r\rightarrow \infty }$) is equivalent to taking all ${\displaystyle \mathbf {x} _{i}}$ to zero, and by changing coordinates back to r, ${\displaystyle \theta }$ and ${\displaystyle \phi }$, and redefining ${\displaystyle r\rightarrow r/{\sqrt {n}}}$, we get the asymptotic metric

${\displaystyle ds^{2}=dr^{2}+{\frac {r^{2}}{4}}\left({d\psi \over n}+\cos \theta \,d\phi \right)^{2}+{\frac {r^{2}}{4}}[(\sigma _{1}^{L})^{2}+(\sigma _{2}^{L})^{2}].}$

This is R⁴/Z_n = C²/Z_n, because it is R⁴ with the angular coordinate ${\displaystyle \psi }$ replaced by ${\displaystyle \psi /n}$, which has the wrong periodicity (${\displaystyle 4\pi /n}$ instead of ${\displaystyle 4\pi }$). In other words, it is R⁴ identified under ${\displaystyle \psi \ {\sim }\ \psi +4\pi k/n}$, or, equivalently, C² identified under z_i ~ ${\displaystyle e^{2\pi ik/n}}$ z_i for i = 1, 2.

To conclude, the multi-center Eguchi–Hanson geometry is a Kähler Ricci flat geometry which is asymptotically C²/Z_n. According to Yau's theorem this is the only geometry satisfying these properties. Therefore, this is also the geometry of a C²/Z_n orbifold in string theory after its conical singularity has been smoothed away by its "blow up" (i.e., deformation).^[3]

Gibbons–Hawking multi-centre metrics

The Gibbons–Hawking multi-center metrics are given by^[4][5]

${\displaystyle ds^{2}={\frac {1}{V(\mathbf {x} )}}(d\tau +{\boldsymbol {\omega }}\cdot d\mathbf {x} )^{2}+V(\mathbf {x} )d\mathbf {x} \cdot d\mathbf {x} ,}$

where

${\displaystyle \nabla V=\pm \nabla \times {\boldsymbol {\omega }},\quad V=\varepsilon +2M\sum _{i=1}^{k}{\frac {1}{|\mathbf {x} -\mathbf {x} _{i}|}}.}$

Here, ${\displaystyle \epsilon =1}$ corresponds to multi-Taub–NUT, ${\displaystyle \epsilon =0}$ and ${\displaystyle k=1}$ is flat space, and ${\displaystyle \epsilon =0}$ and ${\displaystyle k=2}$ is the Eguchi–Hanson solution (in different coordinates).

FLRW-metrics as gravitational instantons

In 2021 it was found ^[6] that if one views the curvature parameter of a foliated maximally symmetric space as a continues function, the gravitational action, as a sum of the Einstein–Hilbert action and the Gibbons–Hawking–York boundary term, becomes that of a point particle. Then the trajectory is the scale factor and the curvature parameter is viewed as the potential. For the solutions restricted like this general relativity takes the form of a topological Yang–Mills theory.