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Let's suppose I am an ant who lives in a 2D curved space. Locally the world seems 2d-euclidean to me, but it is not if I consider a large portion of space.

Now let's consider a human being who lives in curved 4d spacetime. Locally the spacetime seems 4d-euclidean --> in fact, the space itself appears 3d-euclidean to us. But if we consider a larger portion of spacetime, then the curvature will inevitably be evident.

However: if I look around me in any moment, the space seems to me 3d euclidean EVERYWHERE, not only locally. I mean, it seems like I can extend my 3d-euclidean-grid forever in space (at least according to my eyes).

How can this be possible if spacetime is curved? Do our eyes give us misleading impressions?

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    $\begingroup$ how would our eyes see light following the curvature of space as not being straight $\endgroup$ – Adrian Howard Sep 9 at 11:41
  • $\begingroup$ @adrian I don't know. But what our eyes see is an euclidean 3d flat space even though it is not... do you agree? $\endgroup$ – Federico Toso Sep 9 at 11:52
  • $\begingroup$ My eyes currently tell me that there is no universe beyond my living room. Should I believe them? $\endgroup$ – WillO Sep 9 at 11:58
  • $\begingroup$ Maybe read Kant $\endgroup$ – Stéphane Rollandin Sep 9 at 12:36
  • $\begingroup$ Consider to define the word Euclidean. It has different meanings in the litterature. Do you mean flat? $\endgroup$ – Qmechanic Sep 9 at 15:42
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You need sensitive instruments/telescopes to see that light does not always follow straight lines, but it is possible - see Tests of General Relativity.

In the everyday world, if I want to send an apple from A so that it arrives 1 second later at B 1 metre away, then I have to throw it so that it follows a ballistic trajectory. If I just aim it in a straight line from A to B at a velocity of 1 metre per second then it will fall to the floor. This is actually evidence that our local 4D spacetime is curved by the nearby presence of a large mass (the Earth). But we don't intuitively think of it that way.

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