B Are the 3 space dimensions "equal"?

[Gerinski]

The question originates from what is often said in accounts of superstring theory, that (perhaps) 7 space dimensions might be curled up into such a tiny scale as to become undetectable.

So the question came up, if dimensions might be curled up or extended into different scales, say "very tight" or "very open", might that mean that also our 3 familiar space dimensions could be extended into different scales? and if so, what might that mean in practice, if anything?

We assume that the 3 space dimensions of our spacetime are "equally extended", the 3 of them to the same scale. 1 light year has the same extension in any direction or at least it looks so to us. Is it conceivable in any way that the 3 spatial dimensions might be extended to different extents in our whole universe? and if so, what might that mean? could there be any observational clues in one or another sense?

Thanks!

 

[mathman]

On a large scale, the universe is isotropic, so all three dimensions have the same properties.

 

[Gerinski]

Thanks. Assuming hypothetically that the 3 spatial dimensions might have extended with different rates, might that have any observational clues in the present time? Of course our observable universe extends far enough in any of the 3 dimensions so all 3 of them extend beyond our observable universe. But for example in large-scale geometrical measurements, might we (hypothetically) detect that "a cube" does not really have precisely the same extension in the 3 dimensions? or on the contrary, just because we measure the cube as that region with equal extension in the 3 dimensions, it would be impossible for us to detect any discrepancy in the extension of the 3 dimensions?
I am aware that the question may sound a but silly, sorry if that's the case...

 

[kimbyd]

Thanks. Assuming hypothetically that the 3 spatial dimensions might have extended with different rates, might that have any observational clues in the present time? Of course our observable universe extends far enough in any of the 3 dimensions so all 3 of them extend beyond our observable universe. But for example in large-scale geometrical measurements, might we (hypothetically) detect that "a cube" does not really have precisely the same extension in the 3 dimensions? or on the contrary, just because we measure the cube as that region with equal extension in the 3 dimensions, it would be impossible for us to detect any discrepancy in the extension of the 3 dimensions?
I am aware that the question may sound a but silly, sorry if that's the case...

This kind of thing has been investigated for inflation, but there isn't any observational evidence of it as yet. Here's one example model:
https://arxiv.org/abs/1511.01683

The above theoretical idea was based upon some potential anisotropies observed in the WMAP data. But those anisotropies aren't clearly distinguishable from random noise.

So the answer is that yes, this kind of idea has been considered and investigated, and so far there isn't any clear evidence of anything like it happening.

 

[newjerseyrunner]

I am aware that the question may sound a but silly, sorry if that's the case...

These are not silly questions, it's part of the deep mystery of what is space itself? Why are there three? Are they exactly the same? Why is there a time dimension that behaves completely differently?

 

[kurros]

Hmm, this would imply some kind of violation of Lorentz invariance wouldn't it? I mean Lorentz invariance doesn't apply over compactified dimensions, so i assume it would also get screwed up in some large yet still compact dimensions, unless they were all compact "together" so to speak, like in de Sitter spacetime.

 

[kimbyd]

Hmm, this would imply some kind of violation of Lorentz invariance wouldn't it? I mean Lorentz invariance doesn't apply over compactified dimensions, so i assume it would also get screwed up in some large yet still compact dimensions, unless they were all compact "together" so to speak, like in de Sitter spacetime.

No more than General Relativity already violates Lorentz invariance, as Lorentz invariance in its simplest form doesn't take into account space-time curvature.

 

[JMz]

No more than General Relativity already violates Lorentz invariance, as Lorentz invariance in its simplest form doesn't take into account space-time curvature.

Conversely, GR allows (in fact, requires) local Lorentz invariance, and this would still be true if the 3 spatial dimensions were unequally "large": say, 1 compactified though very large, the other 2 non-compact. Locally, we measure a very high degree of isotropy, so Lorentz is appropriate, whether or not something is different at the scale of, say, 10^12 LY or more.

 

[kurros]

Hmm I guess that's true. So we are actually talking about some bizarre deformation of de Sitter spacetime? Like some Lorentzian n-ellipsoid rather than an n-sphere?

 

[JMz]

Hmm I guess that's true. So we are actually talking about some bizarre deformation of de Sitter spacetime? Like some Lorentzian n-ellipsoid rather than an n-sphere?

Well, I would not say it that way: The "ellipsoid" suggests to me that the distances would have a local (spatial, inertial) metric that ia not Euclidean. That is not what I meant, and observationally, that it not supported. What I was thinking of was the possibility that one or more of the dimensions might be "circular", so that the 3-space might be a cylinder, torus, or 3-sphere, albeit extremely large.

 

[kurros]

Well, I would not say it that way: The "ellipsoid" suggests to me that the distances would have a local (spatial, inertial) metric that ia not Euclidean. That is not what I meant, and observationally, that it not supported. What I was thinking of was the possibility that one or more of the dimensions might be "circular", so that the 3-space might be a cylinder, torus, or 3-sphere, albeit extremely large.

Oh yeah I assumed the ellipsoid would be vast such that you would not notice anything. Except perhaps for some anisotropy in the accelerated expansion of the universe maybe? I'm not sure what happens to the cosmological constant in a spacetime that is less symmetric than dS. But ok I guess you could also have some flat spacetime that is topologically a cylinder or torus or something. I guess my original comment about Lorentz invariance was really just a poorly-phrased concern about whether such a spacetime is allowed by General Relativity. Can you really have such a thing in GR? I have some intuition that it must violate something important, though I cannot back that up with anything.

 

[kimbyd]

Conversely, GR allows (in fact, requires) local Lorentz invariance, and this would still be true if the 3 spatial dimensions were unequally "large": say, 1 compactified though very large, the other 2 non-compact. Locally, we measure a very high degree of isotropy, so Lorentz is appropriate, whether or not something is different at the scale of, say, 10^12 LY or more.

Right. Another way to state it is that General Relativity provides a more generalized form of Lorentz invariance. Having different spatial dimensions have different properties is a possibility well-described by General Relativity, and thus wouldn't violate the more generalized form of Lorentz invariance described by the theory.