- Gravitational instanton
In
mathematical physics anddifferential geometry , a gravitational instanton is a four-dimensional completeRiemannian manifold satisfying thevacuum Einstein equation s. They are so named because they are analogues in quantum theories of gravity ofinstanton s inYang–Mills theory . In accordance with this analogy with self-dual Yang–Mills instantons, gravitational instantons are usually assumed to look like four dimensionalEuclidean space at large distances, and to have self-dualRiemann 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 ofEinstein manifold s. From a physical point of view, a gravitational instanton is a non-singular solution of the vacuumEinstein equation s 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 thehyperkähler quotient construction.Properties
* A four-dimensional Kähler-
Einstein manifold has a self-dualRiemann tensor .
* Equivalently, a self-dual gravitational instanton is a four-dimensional completehyperkähler manifold .
* Gravitational instantons are analogous to self-dual Yang-Mills instantons.Taxonomy
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). There also exist ALG spaces whose name is chosen by induction.
Examples
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 ofEuler angles by:sigma_1 = sin phi , d heta - cos phi sin heta , d phi:sigma_2 = cos phi , d heta + sin phi sin heta , d phi:sigma_3 = d psi + cos heta , d phi
Taub–NUT metric
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
ds^2 = left( 1 - frac{a}{r^4} ight) ^{-1} dr^2 + frac{r^2}{4} left( 1 - frac{a}{r^4} ight) {sigma_3}^2 + frac{r^2}{4} ({sigma_1}^2 + {sigma_2}^2).
where r ge a^{1/4}.This metric is smooth everywhere if it has no conical singularity at r ightarrow a^{1/4}, heta = 0, pi. For a = 0 this happens if psi has a period of 4pi, which gives a flat metric on R4; However for a e 0 this happens if psi has a period of 2pi.
Asymptotically (i.e., in the limit r ightarrow infty) the metric looks like: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 a e 0, psi has only half the usual periodicity, as we have seen. Thus the metric is asymptotically R4 with the identification psi ~ psi + 2pi, which is a Z2
subgroup ofSO(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:ds^2 = frac{1}{V(mathbf{x})} ( d psi + oldsymbol{omega} cdot d mathbf{x})^2 + V(mathbf{x}) d mathbf{x} cdot d mathbf{x},whereabla V = pm abla imes oldsymbol{omega}, quad V = sum_{i=1}^2 frac{1}.For some "n" points 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 (r ightarrow infty) is equivalent to taking all mathbf{x}_i to zero, and by changing coordinates back to r, heta and phi, and redefining r ightarrow r/sqrt{n}, we get the asymptotic metric :ds^2 = dr^2 + frac{r^2}{4} ({dpsiover n} + cos heta dphi)^2 + frac{r^2}{4} [(sigma_1^L)^2 + (sigma_2^L)^2] . This is R4/Zn = C2/Zn, because it is R4 with the angular coordinate psi replaced by psi/n, which has the wrong periodicity (4pi/n instead of 4pi). In other words, it is R4 identified under psi ~ psi + 4pi k/n, or, equivalnetly, C2 identified under zi ~ e^{2pi i k/n} 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 instring theory after its conical singularity has been smoothed away by its "blow up" (i.e., deformation) [http://arxiv.org/abs/hep-th/9603167] .Gibbons–Hawking multi-centre metrics
ds^2 = frac{1}{V(mathbf{x})} ( d au + oldsymbol{omega} cdot d mathbf{x})^2 + V(mathbf{x}) d mathbf{x} cdot d mathbf{x},
where
abla V = pm abla imes oldsymbol{omega}, quad V = varepsilon + 2M sum_{i=1}^{k} frac{1}{|mathbf{x} - mathbf{x}_i | }.
epsilon = 1 corresponds to multi-Taub–NUT, epsilon = 0 and k = 1 is flat space, and epsilon = 0 and k = 2 is the Eguchi–Hanson solution (in different coordinates).
See also
*
Hyperkähler manifold References
* Gibbons, G. W.; Hawking, S. W., "Gravitational Multi-instantons". Phys. Lett. B 78 (1978), no. 4, 430--432; see also "Classification of gravitational instanton symmetries". Comm. Math. Phys. 66 (1979), no. 3, 291--310.
* Eguchi, Tohru; Hanson, Andrew J., "Asymptotically flat selfdual solutions to Euclidean gravity". Phys. Lett. B 74 (1978), no. 3, 249--251; see also "Self-dual solutions to Euclidean Gravity". Ann. Physics 120 (1979), no. 1, 82--106 and "Gravitational instantons". Gen. Relativity Gravitation 11 (1979), no. 5, 315--320.
* Kronheimer, P. B., "The construction of ALE spaces as hyper-Kähler quotients". J. Differential Geom. 29 (1989), no. 3, 665--683.
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