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 (or perhaps asymptotically locally flat) 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.
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 (ALE spaces), asymptotically locally flat spaces (ALF 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 Kähler manifolds; and various characteristic classes, such as Euler characteristic, the Hirzebruch signature (Pontryagin class), the Rarita–Schwinger index (spin-3/2 index), or generally the Chern class. The ability to support a spin structure (i.e. to allow consistent Dirac spinors) is another appealing feature.
Eguchi et al. list a number of examples of gravitational instantons.[1] These include, among others:
R4
T4
S4
S2 x S2
R2 x S2
R2 x S2
T*CP(1)
CP(2).
CP(2) ⊕ \overline{CP
CP(2)\setminus\{0\}
L(k+1,1)
L(2,1)
L(2,1)
It will be convenient to write the gravitational instanton solutions below using left-invariant 1-forms on the three-sphere S3 (viewed as the group Sp(1) or SU(2)). These can be defined in terms of Euler angles by
\begin{align} \sigma1&=\sin\psid\theta-\cos\psi\sin\thetad\phi\\ \sigma2&=\cos\psid\theta+\sin\psi\sin\thetad\phi\\ \sigma3&=d\psi+\cos\thetad\phi.\\ \end{align}
Note that
d\sigmai+\sigmaj\wedge\sigmak=0
i,j,k=1,2,3
See main article: Taub–NUT space.
ds2=
| 1 | |
| 4 |
| r+n | |
| r-n |
dr2+
| r-n | |
| r+n |
n2
| 2 | |
| {\sigma | |
| 3} |
+
| 1 | |
| 4 |
(r2-
| 2 | |
| n | |
| 1} |
+
| 2) | |
| {\sigma | |
| 2} |
The Eguchi–Hanson space is defined by a metric the cotangent bundle of the 2-sphere
T*CP(1)=T*S2
ds2=\left(1-
| a | |
| r4 |
\right)-1dr2+
| r2 | |
| 4 |
\left(1-
| a | |
| r4 |
\right)
| 2 | |
| {\sigma | |
| 3} |
+
| r2 | |
| 4 |
| 2 | |
| (\sigma | |
| 1 |
+
| 2). | |
| \sigma | |
| 2 |
where
r\gea1/4
r → a1/4
\theta=0,\pi
a=0
\psi
4\pi
a\ne0
\psi
2\pi
Asymptotically (i.e., in the limit
r → infty
ds2=dr2+
| r2 | |
| 4 |
| 2 | |
| \sigma | |
| 3 |
+
| r2 | |
| 4 |
| 2 | |
| (\sigma | |
| 1 |
+
| 2) | |
| \sigma | |
| 2 |
a\ne0
\psi
\psi{\sim}\psi+2\pi
There is a transformation to another coordinate system, in which the metric looks like
ds2=
| 1 | |
| V(x) |
(d\psi+\boldsymbol{\omega} ⋅ dx)2+V(x)dx ⋅ dx,
\nablaV=\pm\nabla x \boldsymbol{\omega}, V=
| 2 | |
| \sum | |
| i=1 |
| 1 | |
| |x-xi| |
.
(For a = 0,
V=
| 1 | |
| |x| |
\rho=r2/4
\rho
\theta
\phi
x
x=(\rho\sin\theta\cos\phi,\rho\sin\theta\sin\phi,\rho\cos\theta)
In the new coordinates,
\psi
\psi {\sim} \psi+4\pi.
One may replace V by
V=
| n | |
| \sum | |
| i=1 |
| 1 | |
| |x-xi| |
.
xi
r → infty
xi
\theta
\phi
r → r/\sqrt{n}
ds2=dr2+
| r2 | |
| 4 |
\left({d\psi\overn}+\cos\thetad\phi\right)2+
| r2 | |
| 4 |
| L) | |
| [(\sigma | |
| 1 |
2+
| L) | |
| (\sigma | |
| 2 |
2].
This is R4/Zn = C2/Zn, because it is R4 with the angular coordinate
\psi
\psi/n
4\pi/n
4\pi
\psi {\sim} \psi+4\pik/n
e2\pi
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" (i.e., deformation).[3]
The Gibbons–Hawking multi-center metrics are given by[4] [5]
ds2=
| 1 | |
| V(x) |
(d\tau+\boldsymbol{\omega} ⋅ dx)2+V(x)dx ⋅ dx,
where
\nablaV=\pm\nabla x \boldsymbol{\omega}, V=\varepsilon+2M
| k | |
| \sum | |
| i=1 |
| 1 | |
| |x-xi| |
.
Here,
\epsilon=1
\epsilon=0
k=1
\epsilon=0
k=2
In 2021 it was found[6] that if one views the curvature parameter of a foliated maximally symmetric space as a continuous 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.
k(\phi)