this post was submitted on 02 Oct 2023
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What do you mean by that?
Look at a piece of paper. Mark two points on it that are a good distance apart. Travel can only happen along the surface of the paper. When it's flat, your time of travel depends on the distance between the two points on that paper.
Now dap a spot of glue next to one of the points and fold the paper in such a way that the other point comes to rest close to that glue. Wait for it to bind and then spread the paper a little without breaking the glue. That glued point is a wormhole, a place where two points of that flat 2D universe touch despite not being next to each other. Travel from point A to point B is now a shorter distance thanks to the wormhole. But there is no way in which the paper universe can be described as flat anymore.
Or think of a papermache ball, that's also made from paper but if you travel long enough in one direction, you'll end up where you started. Because it isn't flat.
Now our universe is 3D not 2D, but from a higher dimensional perspective it has the same prooerties of flatness as that 2D paper has for us.
Geometrically speaking, if you draw a really, really big triangle between, say three galaxies, the angles of the triangle add to 180° in a flat universe. In a non-flat space, this would not be the case. For example, if you draw a triangle between, say, New York, Berlin and Rio de Janeiro on the surface of the Earth, the three angles between the lines would add to more than 180° since Earth is topologically a sphere and not flat. And if you draw three lines beween three points on a saddle shape like a Pringles chip, you'll find that the angles add up to less than 180°.
Fun fact: topologically speaking, no matter how you fold or bend a sheet of paper, it remains flat. A cilinder is a flat surface with zero curvature!
What I mean is that what we observe is best approximated as a relative lack of spatial curvature over long distance scales. One of the lines of evidence for this is the cosmic microwave background. A flat topology for space-time over long distances would look different than an open or closed universe: https://phys.org/news/2017-06-universe-flat-topology.html