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As I know, GR does not need to assume anything about a >3D space, where our 3+1 spacetime can be embedded into. However, I think the curved 3D space (more clearly, the spacelike cuts of the 3+1D spacetime) can be embedded into a higher-dimensional space, only GR can not say anything about it.

Is it known, at least how many dimensions should a flat, "super-space" have?

Qmechanic
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peterh
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    Does this answer your question? Can a non-Euclidean space be descripted through an Euclidean space of higher dimension? So why use non-Euclidean? – John Rennie Nov 13 '23 at 20:08
  • @JohnRennie The back & forth in the post you referred to just begs the question if there is a "reasonably simple" sufficiency criterion on an m-dim manifold that ensures its embedding in an m+1 or m+2, etc. dimensional flat space. Is there one? Admittedly, I do not know how to define "reasonably simple", but it is like some other things, as Justice Stewart stated of pornography "I know it when I see it". – hyportnex Nov 14 '23 at 00:21
  • @JohnRennie I believe the answer is false, it gives irrationally high results. For example, embedding a curved 1D line clearly does not need 17 dimensions. It needs 2. – peterh Nov 14 '23 at 06:34
  • embedding a curved 1D line clearly does not need 17 dimensions. It needs 2. This is a good example of why intuition can let use down. A trefoil knot is a curved 1D line but cannot be embedded into 2D space because it would self intersect. You need 3D. How confident are you that there are no shapes drawn with a 1D line that need more than 3D? – John Rennie Nov 14 '23 at 06:46
  • @JohnRennie I believe, self-intersections should be allowed here. – peterh Nov 14 '23 at 07:08
  • Then you believe wrongly. – John Rennie Nov 14 '23 at 07:09
  • @JohnRennie Why? It would still preserve length. – peterh Nov 14 '23 at 07:10
  • @JohnRennie Btw, imagine if there is a much smaller D euclidean world, in which our spacetime is embedded! If we allow self-intersections, probably already 6D or not much more is enough to embed our curved 3D space. Now next interesting thing would be, to create such a curvature where the intersections happen as large overlaps... maybe... maybe... that would lead to quite interesting, sci-fi phenomena. – peterh Nov 14 '23 at 12:31

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