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According this answer, the recent WMAP experiment has only shown that if our Universe has a spherical geometry, then it should have at least a $3\cdot 10^{11}$ light year big radius.

Now consider the possibility, if our Universe is a 4-sphere, thus it has a small, constant positive curvature.

It means, that we have a new symmetry. Translating any point of any system with $2{\pi}r$, we get the same system back. Note, it is a different thing as the common space translation symmetry (which results the impulse preservation):

  1. it is valid only for $2{\pi}r$ translations
  2. but, it is valid for any point of any system, not only for the whole system.

On Noether's theorem, every differentiable symmetry of an action has a corresponding conservation law.

What conservation law would correspond to this symmetry?

Extension/Fix:

As I understand @conifold 's answer, this is a discrete and not a continuous symmetry, because the translation is possible here only with $n\cdot 2\pi r (n \in \mathbb{Z})$, thus Noether's theorem here doesn't apply in its original form. But, according to this question, yes, there is something similar to Noether's theorem also on discrete symmetries. On the accepted (and bountied) answer, "For infinite symmetries like lattice translations the conserved quantity is continuous, albeit a periodic one." How does this apply in our case?

Community
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peterh
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Your "translation of any point of any system with 2πr" can not be done for all points of the sphere simultaneously. Therefore it is not a symmetry in the sense of the Noether's theorem. I am guessing it refers to something like the full rotation of a 2-sphere around an axis, and you can already see from this example that you can not perform such a rotation on all points at once. Some move the full circle, others less, some not at all (poles). For the 3-sphere there may not be poles, but then there will be invariant circles, for the 4-sphere there will be poles again (this follows from existence of 1D or 2D invariant subspaces in real linear algebra).

But "the shape of the universe" being a sphere refers to a spacetime slice, not the whole spacetime, so it is a 3-sphere. It would be problematic for a 4D spacetime to be spherical even on the cyclic cosmology theories. They are also probing WMAP data for detecting other finite 3D space-forms, the quotients of the sphere by finite groups, see The Poincaré Dodecahedral Space and the Mystery of the Missing Fluctuations by Weeks.

Even if it did work globally "translation by 2πr" has no continuous parameter in it (r is fixed), so the Noether's theorem would still not apply. However, there is a shadow of it for discrete symmetries, involving conserved topological charges, which impose selection rules on various processes, see Is there something similar to Noether's theorem for discrete symmetries?

Conifold
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I am truly not an expert on this, but my superficial impression is that preservation theorems correspond to local (infinitesimal) symmetries, while you are talking about a (slightly doubtful) global symmetry? (Doubtful insofar as you would assume a perfectly constant curvature, which seems a rather unphysical assumption?)

Jakob
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In short, this symmetry isn't a differentiable symmetry in the sense as it would be affected by the Noether Theorem.

peterh
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