Is the folding sheet analogy really that good for understanding what a wormhole is? After all, space-time curvature doesn't require any ambient space (it's intrinsic), as such a picture would suggest. Also the analogy is only about the spatial part, but maybe that's just because it would be hard to visualize otherwise. Is there a better visual analogy?
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$\begingroup$ Generally, it is not a good analogy (makes use of an ambient space, adds a folding curvature somewhere else, usually involves some weird pencil stabbing through sheet...). But when looking for a better analogy we need to specify what aspects are to be conveyed - it is hard to find good analogs for even simple metrics (Flamm's paraboloid for the spatial part of the Schwarzschild metric is surprisingly confusing). $\endgroup$– Anders SandbergCommented May 29 at 7:14
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$\begingroup$ Related and possibly a duplicate: Are wormholes really a valid shortcut to distant points in the universe? $\endgroup$– John RennieCommented May 29 at 7:22
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1$\begingroup$ This answer of mine is also relevant, even though it's on the science fiction SE :-) $\endgroup$– John RennieCommented May 29 at 7:24
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Unfortunately the folding paper is all you’re really going to get.
If you were to embed our four-dimensional spacetime in something of higher dimensions, you would be able to see a definite “wormhole”-shaped thing through which you could draw curves that light or people could travel across. The issue is, humans are meant to think in at most three dimensions, not five. What the piece of paper does is “projects” our 4D universe onto a 2D sheet. Because the 2D sheet is embedded in 3(+1)D spacetime, you can fold it and cut it and draw on it and conveniently imagine what a wormhole would look like on it.
More-accurate analogies would involve “folding” 3D objects. Unless you can figure out how to do that, you’re not gonna have much luck.
What you might be able to do is imagine a sort of lens-shaped object that makes objects appear much larger than they are from one angle, except when you go through the object, they stay that size - because you travelled a much larger distance than expected, and now they just are that size according to your eyes because they’re way closer to you.
As far as accuracy, folding paper actually is pretty close. What a wormhole does is create a region of spacetime where the distance between two points is just less, allowing a constant-velocity observer to cross the space between those points in much less time than would ordinarily be allowed. If spacetime were 2D, then one very accurate way to visualize a wormhole would be to just bring two points on the sheet closer to each other by folding the sheet halfway; crossing the distance while folded is intuitively much faster than crossing the distance all the way across the flat sheet.
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$\begingroup$ In your paragraph where you talk about higher dimensions, are you saying the same thing as this?: “Many scientists postulate that wormholes are merely projections of a fourth spatial dimension, analogous to how a two-dimensional (2D) being could experience only part of a three-dimensional (3D) object.[2] A well-known analogy of such constructs is provided by the Klein bottle, displaying a hole when rendered in three dimensions but not in four or higher dimensions.” $\endgroup$ Commented Jun 3 at 2:28
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$\begingroup$ Sort of??? It's known that the universe is experientially fourth-dimensional, and while we haven't found a working wormhole (to my knowledge), what it would be is essentially a projection of a five-dimensional space onto our regular 4D spacetime that allows two points to come closer together. But you're actually not that far off! $\endgroup$ Commented Jun 3 at 12:25