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Spacetime curvature is given by the cosmological constant, that produces a De-Sitter spacetime. It is non-zero.

But space curvature is nearly zero (how close to zero, compared to the cosmological constant?)

If we accept Kaluza-Klein derived theories (I like to believe that that encompasses string theory and M-theory as well, but one is never sure these days), then one would expect that curvatures of hidden dimensions are huge to be able to stay hidden (by the way, one can conceive high curvature manifolds that are not bounded, hence is not clear how high curvature helps hidden dimensions to stay hidden in all cases)

I'm wondering if one can think of individual dimensional curvatures as cosmological variables, and time was the last one to expand exponentially, while space dimensions where older than time (in some mathematically yet unrealised sense), and electromagnetic gauge $U(1)$ is just the higher inner dimension

Can we compare the curvature of a typical Kaluza-Klein theory compactification with the curvature of the cosmological constant? what about the curvature of the U(1) electromagnetic gauge dimension? can one express the 1/137 coupling constant of electromagnetism as a curvature?

I'm aware that there is no evidence for the electromagnetic dimension to budge or expand in spectroscopic analysis of light from the farthest observed galaxies, but I think that's still an interesting scenario to think about

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    $\begingroup$ I'm not sure what you mean by "curvature of hidden dimensions". Curvature is a property of an entire manifold, not of a "dimension", and there are extra-dimensional models where the metric is not a product metric, so we can't reasonably separate the thing into our four dimensions and the "extra dimension" if we want to calculate the curvature. $\endgroup$ – ACuriousMind Aug 8 '15 at 13:05
  • $\begingroup$ "Curvature is a property of an entire manifold, not of a "dimension"", well, even when the original metric is not a product metric, someone with enough physical intuition could still conceive some sort of cooling process/phase transition that induced a preferred direction in the manifold, that would expand tightly while some other preferred directions would stay unaffected (either expanded or curled) $\endgroup$ – diffeomorphism Aug 31 '15 at 19:05
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Can you compare curvatures in space, spacetime

Yes, you can compare spatial curvature with spacetime curvature. Have a look at this Baez article: "Similarly, in general relativity gravity is not really a 'force', but just a manifestation of the curvature of spacetime. Note: not the curvature of space, but of spacetime. The distinction is crucial". There's sometimes some confusion concerning curved spacetime and curved space. Curved spacetime is not curved space and curved time. It's a curvature of "the metric". A curvature in your plot of measurements, as it were.

and hidden space?

I don't know anything about hidden space I'm afraid.

Spacetime curvature is given by the cosmological constant, that produces a De-Sitter spacetime. It is non-zero.

Spacetime curvature usually relates to the tidal force and the second derivative of gravitational potential. Have a google on that. Then check out the De Sitter universe on Wikipedia where you can read that it "models the universe as spatially flat". There's room for confusion here. I'll try to clarify...

But space curvature is nearly zero (how close to zero, compared to the cosmological constant?)

See Einstein talking about a gravitational field as space that's "neither homogeneous nor isotropic". Then see this paper which says curved spacetime is the same thing as inhomogeneous space. Then note how the FLRW metric starts with "the assumption of homogeneity and isotropy of space". That's a reasonable assumption wherein we might claim there's no overall gravitational field in the universe, because it didn't collapse when it was small and dense. Note that spacetime is curved because the universe is expanding, but that this isn't the same as the spacetime curvature of a gravitational field. In the former situation space is inhomogeneous over time, in the latter it's inhomogeneous across space.

If we accept Kaluza-Klein derived theories (I like to believe that that encompasses string theory and M-theory as well, but one is never sure these days), then one would expect that curvatures of hidden dimensions are huge to be able to stay hidden

There's no supporting evidence for string theory or M theory or any hidden dimensions.

Can we compare the curvature of a typical Kaluza-Klein theory compactification with the curvature of the cosmological constant?

I would say no, because they're chalk and cheese.

what about the curvature of the U(1) electromagnetic gauge dimension?

I would encourage you to pursue this. Check out electromagnetic geometry and things like this by Percy Hammond: "We conclude that the field describes the curvature that characterizes the electromagnetic interaction". For an analogy, imagine you're standing on a headland overlooking a flat calm sea near an estuary. You see a single wave, and you notice that its path curves a little. That's because there's a salinity gradient in the water, which is inhomogeneous. The wave is standing in for a photon, and the inhomogeneous water is standing in for a gravitational field. Now look at the surface of the sea where the wave is. It's curved.

can one express the 1/137 coupling constant of electromagnetism as a curvature?

Not directly, because it's running constant. See NIST. At an energy corresponding to the mass of the W boson : (mW) is approximately 1/128 compared with its zero-energy value of approximately 1/137. Thus the famous number 1/137 is not unique or especially fundamental".

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