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Let us assume a universe that is mostly flat on medium and large scales. Suppose we have exotic matter and could make a wormhole. Then you move one of the entrances to the wormhole very far away. Now you have two distant regions of a flat universe connected trough a shortcut, a wormhole. But how is this possible? We can always imagine a spacetime as embedded into a spacetime of additional dimensions, for instance, a closed 2D universe of constant curvature could be imagined as a sphere embedded inside a flat 3D Euclidean space (of course, it does not mean that extra dimension exists).

Assuming what I said until now is correct: how can you imagine that you can connect two distant point of a flat spacetime trough shortcuts? any "tunnels" must surely be longer that the actual distance in the original space, which is a pretty straight line. That is, we usually visualize wormholes as a shorter path within spacetime, and usually we depict this by folding a flat sheet and connecting two distant point on the sheet through a wormhole shortcut, like in the image below. But what if the space is not initially folded?

PS: inspired in a comment below by AVLabs. I see, but if that is the case, then another question comes to my mind. Let us assume there is some initial flat spacetime, embedded in a higher dimension. It is natural to think that the actual shape, even if it could be anything compatible with flat, is a specific instance. It makes no sense to me that such space will globally change shapes dynamically depending on where do I make wormholes. Thus, it seems natural to conclude the the space of GR is NOT embedded in extra dimensions, because if this were the case the shape of the entire GR surface would need to fold to adapt to the wormholes of my choice. enter image description here

  • Possibly related to this answer, which explains that even a cylinder is intrinsically ''flat". On the sheet, you wouldn't be able to measure the sheet is actually shaped as in the picture. – A-V Labs Jul 02 '22 at 20:49
  • @A-VLabs Thanks! That actually triggered a follow up question that I added to the end. –  Jul 02 '22 at 21:18
  • The image is somewhat misleading. Rather than thinking of the "shortcut" as going through some ambient higher dimensional space, think of going through a teleport, like in the Portal video game. There's no ambient space, it's just a visualization aid - the teleport makes a topological identification with a region somewhere else in the universe, effectively changing the topology of the spacetime you inhabit (redefining what constitutes the neighborhood of each point). – Filip Milovanović Jul 02 '22 at 21:19
  • @FilipMilovanović I see, so would that be a good argument against the existence of an actual embedding space? or is it irrelevant? –  Jul 02 '22 at 21:34
  • The best argument against an embedding space is it creates a infinite regress and adds no explanatory or predictive power. If you explain the metric of this manifold as induced by the metric of another manifold, of greater dimension and probably not Minkowski, why stop there? – J.G. Jul 02 '22 at 21:37
  • @J.G. yes, makes sense, thanks! –  Jul 02 '22 at 21:39
  • "would that be a good argument against the existence of an actual embedding space" - if I'm not mistaken, it doesn't by itself exclude the possibility of an embedding space, it just doesn't require one. As J.G. said - it's basically the Occam's razor type of argument - unless we encounter some set of phenomena that cannot really be explained elegantly without an embedding space (or otherwise, unless we find some sort of evidence for an embedding space), there's no reason to postulate one. – Filip Milovanović Jul 03 '22 at 21:39

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