Comparing Vacuum Solutions to Topologies of Einstein Space

[TGlad]

Hi,

Wikipedia lists about 10 vacuum solutions of the Einstein Field Equations.
However, if I look for topologies of Einstein space, there are many different families, which include Calabi-Yau manifolds, of which abelian, Enriques, Hyperelliptic and K3 surfaces are subsets. Within K3 surfaces alone there are computed lists of 15,000 families (https://projecteuclid.org/euclid.em/1175789798).

So I am wondering how there can be so many known topologies which allow Ricci-flat metrics, but so few known Ricci-flat metrics (vacuum solutions).

I imagine that many topologies don't have analytic "exact" metric formulae, but then I wonder how they can know a topology can admit a Ricci-flat metric if they have no examples of such metrics.

 

[PAllen]

Calubi Yau manifolds may be considered as even dimensional Riemannian manifolds, typically 6. This has little to do with Ricci flat pseudoriemannian manifolds of 4 dimensions. It seems you are comparing totally unlike objects.

 

[TGlad]

Calubi Yau manifolds may be considered as even dimensional Riemannian manifolds, typically 6. This has little to do with Ricci flat pseudoriemannian manifolds of 4 dimensions. It seems you are comparing totally unlike objects.

The wikipedia page says:
"Simple examples of Einstein manifolds include:
Calabi–Yau manifolds admit an Einstein metric that is also Kähler, with Einstein constant {k=0}."

I understand your comment that Calabi-Yau manifolds seem to always be described as Riemannian (rather than pseudo-Riemannian) manifolds.

Aha, maybe this is the source of my confusion. Under the definition of Einstein manifold it says:

"both the dimension and the signature of the metric can be arbitrary, thus not being restricted to the four-dimensional Lorentzian manifolds usually studied in general relativity"
I didn't fully appreciate this. Therefore the statement above that "Calabi–Yau manifolds admit an Einstein metric" does not mean that Calabi-Yau manifolds admit a Lorentzian Ricci-flat metric. And the statement in the Calabi–Yau manifolds page that it "is a particular type of manifold which has properties, such as Ricci flatness" again doesn't mean that it is Lorentzian and Ricci flat.

Does this sound correct?

 

[martinbn]

You also need to keep in mind that some of the existence proofs are not constructive, so you can know that there are many but have less examples. Also I think when people say Einstein metric it is not necessarily related to GR, the definitions is that the Ricci tensor is proportional to the metric tensor, as special case Ricci flat, and no restriction on the dimension nor the signature.