Gravitational instanton


In mathematical physics and differential geometry, a gravitational instanton is a four-dimensional complete Riemannian manifold satisfying the vacuum Einstein equations. They are so named because they are analogues in quantum theories of gravity of instantons in Yang-Mills theory. In accordance with this analogy with self-dual Yang-Mills instantons, gravitational instantons are usually assumed to look like four dimensional Euclidean space at large distances, and to have a self-dual Riemann tensor. Mathematically, this means that they are asymptotically locally Euclidean hyperkähler 4-manifolds, and in this sense, they are special examples of Einstein manifolds. From a physical point of view, a gravitational instanton is a non-singular solution of the vacuum Einstein equations with positive-definite, as opposed to Lorentzian, metric.
There are many possible generalizations of the original conception of a gravitational instanton: for example one can allow gravitational instantons to have a nonzero cosmological constant or a Riemann tensor which is not self-dual. One can also relax the boundary condition that the metric is asymptotically Euclidean.
There are many methods for constructing gravitational instantons, including the Gibbons-Hawking Ansatz, twistor theory, and the hyperkähler quotient construction.

Introduction

Gravitational instantons are interesting, as they offer insights into the quantization of gravity. For example, positive definite asymptotically locally Euclidean metrics are needed as they obey the positive-action conjecture; actions that are unbounded below create divergence in the quantum path integral.
Several distinctions can be made with respect to the structure of the Riemann curvature tensor, pertaining to flatness and self-duality. These include:
By specifying the 'boundary conditions', i.e. the asymptotics of the metric 'at infinity' on a noncompact Riemannian manifold, gravitational instantons are divided into a few classes, such as asymptotically locally Euclidean spaces, asymptotically locally flat spaces.
They can be further characterized by whether the Riemann tensor is self-dual, whether the Weyl tensor is self-dual, or neither; whether or not they are Kahler manifolds; and various characteristic classes, such as Euler characteristic, the Hirzebruch signature, the Rarita—Schwinger index, or generally the Chern class. The ability to support a spin structure is another appealing feature.

List of examples

Eguchi et al. list a number of examples of gravitational instantons. These include, among others:
This is an incomplete list; there are others.

Examples

It will be convenient to write the gravitational instanton solutions below using left-invariant 1-forms on the three-sphere S3 or SU). These can be defined in terms of Euler angles by
Note that for cyclic.

Taub–NUT metric

Eguchi–Hanson metric

The Eguchi–Hanson space is defined by a metric the cotangent bundle of the 2-sphere. This metric is
where. This metric is smooth everywhere if it has no conical singularity at,. For this happens if has a period of, which gives a flat metric on R4; However, for this happens if has a period of.
Asymptotically the metric looks like
which naively seems as the flat metric on R4. However, for, has only half the usual periodicity, as we have seen. Thus the metric is asymptotically R4 with the identification, which is a Z2 subgroup of SO, 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
where
In the new coordinates, has the usual periodicity
One may replace V by
For some n points, 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. The asymptotic limit is equivalent to taking all to zero, and by changing coordinates back to r, and, and redefining, we get the asymptotic metric
This is R4/Zn = C2/Zn, because it is R4 with the angular coordinate replaced by, which has the wrong periodicity. In other words, it is R4 identified under, or, equivalently, C2 identified under zi ~ zi for i = 1, 2.
To conclude, the multi-center Eguchi-Hanson geometry is a Kähler Ricci flat geometry which is asymptotically C2/Zn. According to Yau's theorem this is the only geometry satisfying these properties. Therefore, this is also the geometry of a C2/Zn orbifold in string theory after its conical singularity has been smoothed away by its "blow up".

Gibbons–Hawking multi-centre metrics

The Gibbons-Hawking multi-center metrics are given by
where
Here, corresponds to multi-Taub-NUT, and is flat space, and and is the Eguchi-Hanson solution.