Is Our Understanding of the Space of Einstein Metrics Limited?

[Naty1]

http://arxiv.org/PS_cache/gr-qc/pdf/0310/0310002v3.pdf

...we show that for many spatial topologies, the Hartle-Hawking wave function for a spacetime with a negative cosmological constant develops sharp peaks at certain calculable geometries.

Can someone explain the following statement on page 2 of the paper regarding "our limited understanding of the space of Einstein metrics"...

...for a wide class of manifolds, the sum over topologies produces sharp peaks in the Hartle-Hawking wave function that could not have been guessed by looking at any single contribution. Because of limits to our present understanding of the space of Einstein metrics, a complete, systematic understanding of this phenomenon is still lacking, but ultimately it may be possible to use this sort of analysis to make testable predictions about the geometry and topology of the Universe.

I'm just asking in general, not specifically related to the paper...Is this general statement relative to our universe as well...one a positive cosmological constant?? How is our knowledge limited?? Does this imply a weakness in relativity?


Separately, anyone aware of any papers looking at a possible relationship between the Hawking-Hartle wave function and uncertainty?? Seems like, maybe, if there were wave function peaks [with a positive cosmological constant] they could be related to virtual particles and maybe quantum uncertainty?

Thanks

 

[torquil]

I'm just asking in general, not s...fold"]http://en.wikipedia.org/wiki/4-manifold

 

[Naty1]

More a weakness of a quantum gravity than classical gravity...

ok, that makes sense!