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According to experiments the universe is believed to be flat, meaning that it would follow euclidean geometry. However, is that compatible with the fact that spacetime bends due to gravity? Does euclidean geometry still work when spacetime bends?

I think I'm confusing static universe geometry and spacetime geometry, so I need some explanations. I have a pretty decent background on physics, but very bad general relativity knowledge.

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    $\begingroup$ Note that even flat spacetime in special relativity is described by Minkowski, not Euclidean geometry $\endgroup$
    – fqq
    Commented Sep 17, 2021 at 22:21

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First, the universe is believed to be spatially flat (or close to it); that's not the same as spacetime being flat. The two-dimensional surface of the Earth is curved even though it's a part of a three-dimensional space that is flat (well, close to flat). In a similar way, it's geometrically possible for the three-dimensional surfaces that are called "space" in cosmology to be flat even though the four-dimensional spacetime is curved.

Second, the universe is only approximately spatially flat at very large scales. At small scales, you find all of the local curvature that you would expect from general relativity. This is similar to the Earth's surface appearing to be a perfect sphere (well, oblate spheroid) from a distance, but mountains becoming visible when you zoom in.

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  • $\begingroup$ "Or close to it" is the saving grace in this answer: Dr. Norton, at the U. of Pittsburgh, estimated temporal curvature to be only 10% of the total. Without it, I'd guess that no prediction of recurrent events would be possible. $\endgroup$
    – Edouard
    Commented Sep 27, 2021 at 18:47
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    $\begingroup$ @Edouard I don't know if it was your intention, but calling the phrase "Or close to it" the saving grace in this answer implies that the rest of the answer is bad. On the contrary, this answer is completely correct and precisely addresses the OP's misconception. $\endgroup$
    – J. Murray
    Commented Sep 28, 2021 at 20:36
  • $\begingroup$ Yes, Benrg's answer's completely correct, which is why I upvoted it on the date of my previous comment $\endgroup$
    – Edouard
    Commented Sep 30, 2021 at 15:31
  • $\begingroup$ I had meant "saving grace" to mean something like "most glorious part": As my misuse of English may've cost benrg some increase in his reputation, I'll look thru his recent postings for the sole purpose of seeing if there's anything with which I'm familiar enough to attempt an increase in it, since the OP's question's running about a dozen viewings daily. $\endgroup$
    – Edouard
    Commented Sep 30, 2021 at 15:48
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It looks like you’re mistake is in what a flat universe really means.

While the universe is generally believed to be flat as a large scale structure (meaning, flat in the same way a turbulent ocean looks flat if you zoom out enough) this does not suggest at all that that it behaves in a Euclidean geometric way. Spacetime by its definition is non-Euclidean; even a special relativistic treatment of spacetime shows that pretty quickly.

One indication of this early on in relativistic studies is that while two lines in, let’s say, a Minkowski-spacetime diagram look to be the same length, they are by no means the same length. This is seen in the Minkowski spacetime metric:

(The signs here may flip depending on who you ask)

$$ds^2=dt^2-dx^2-dy^2-dz^2$$

As you can see, this completely negates the Euclidean way of determining lengths that would dictate that all of the signs be positive.

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  • $\begingroup$ Although it was essential in the process of its formulation, Minkowski spacetime is not physical spacetime, because it takes no account of gravity: Consequently, your answer seems incomplete. (A completely flat universe would not only lack any predictability, as per my comment on Benrg's answer, but it would also represent an inefficient storage container--which seems improbable--in terms of the relation between surface and volume.) $\endgroup$
    – Edouard
    Commented Sep 27, 2021 at 19:18
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    $\begingroup$ The mentioning of Minkowski Space here is purely demonstrative; a way of saying “if special relativistic approaches aren’t even Euclidean, general relativistic ones definitively won’t be”. $\endgroup$
    – Justin T
    Commented Sep 28, 2021 at 16:14
  • $\begingroup$ I appreciate that clarification, and admit that it's really the purported flatness of the universe that's more problematic for me: A universe appearing locally (i.e., to us) to be nearly flat might suggest a single universe (or, more probably, a local Universe as in Nikodem Poplawski's model, or a temporal iteration of a single universe, as in Penrose's model) vastly larger than our observable region, which seems plausible to me, but an absolutely flat universe would lack the dynamism of our own, as its components would lock up against each other $\endgroup$
    – Edouard
    Commented Sep 28, 2021 at 17:12
  • $\begingroup$ I agree; I’ll edit to see if I can clarify what I mean by flat! $\endgroup$
    – Justin T
    Commented Sep 28, 2021 at 19:45

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