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From what I understood, a Universe with a positive curvature should behave like a sphere, and light will eventually reach the point where it started. Similarly to a ball, when an ant starts to walk on a ball in a straight line, it will end up where it started.

However, if the Universe is a sphere, we are in the sphere, not on the sphere, right? And if I imagine myself being in the center of a sphere, and I start walking I see no reason I should return to the center.

If you cannot imagine it, this is what I mean. I am in the center, I start to walk on the red line, and what happens if I reach the end?

Enter image description here

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    $\begingroup$ By "sphere" we mean "3-sphere". $\endgroup$ Aug 15, 2021 at 13:39
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    $\begingroup$ "However, if Universe is a sphere, we are in the sphere not on the sphere right?" - No! We are on the surface of a "ball". The part INSIDE the sphere is called a BALL in mathematical terminology. When we say "sphere", we mean "the boundary/surface of a ball". $\endgroup$
    – Prahar
    Aug 15, 2021 at 21:12
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    $\begingroup$ Here's something where you can fly around in a 3-sphere, using WASD to move and click/drag to turn. Ignore the green shooting stars (they're illustrating the Hopf fibration). To give a sense of position, it is showing 24 octahedra assembled into a 24-cell. Where the trusses are lighter blue they're in "your" hemisphere (a 3-ball), and where they become darker they're in the opposite hemisphere (another 3-ball). Far-away dark trusses look big due to lensing from the curvature of space. $\endgroup$ Aug 16, 2021 at 0:30
  • $\begingroup$ Technically we are on the 4 dimensional boundary of a 5 d ball in the de-Sitter picture. But in general, there is no end, you would appear on the other side. You are discussing an "embedded ball", but our universe is not necessarily embedded. 4d ball, or 4d ball in a 5d space, huge difference! $\endgroup$
    – Kregnach
    Aug 16, 2021 at 0:30
  • $\begingroup$ See this answer astronomy.stackexchange.com/questions/10344/… but also, as below, these pictures are 2D analogies - in 3D, only a flat geometry can be pictured, and this doesn't look "flat" by any means; it is simply your good old 3D ("Euclidian") space that you know from your everyday senses. $\endgroup$ Aug 16, 2021 at 3:14

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The thing is, we are not inside a sphere. The 2-dimensional surface of a sphere is just an analogy of how a positively curved space works. It is also just a special case of positively curved space; it has curvature that is both homogeneous and isotropic.

Unfortunately, our brains are not conditioned to think of three dimensions as being curved, either positively or negatively. Instead of trying to draw a picture, the best we can do is to describe such a universe:

  1. A positively curved universe is closed, meaning it is finite in extent but has no edge.
  2. If you walk on a straight line, you will eventually end up where you started from.
  3. The angles of a triangle add up to more than 180 degrees.
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  • $\begingroup$ I have to disagree with the assertion that "our brains are not conditioned to think of three dimensions as being curved, either positively or negatively": They're conditioned to that possibility from conception onward, as neither the womb nor its contents are angular. $\endgroup$
    – Edouard
    Aug 15, 2021 at 19:03
  • $\begingroup$ @Edouard - I think they might have meant conditioned by evolution/genetics, not by environment/etc. If you're a hunter-gatherer trying to sneak up on prey, you're going to be rewarded by successfully modeling the situation in a 3D perspective. $\endgroup$
    – Kevin
    Aug 16, 2021 at 1:15
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    $\begingroup$ @Edouard: walls not being flat or right-angled has nothing to do with the curvature of 3-dimensional space. Our minds do have trouble conceiving of curved 3-space. One way to make a 2-sphere is to take a disc and lift its circular boundary and contract it to a point, giving you a balloon. Analogously, you can make a 3-sphere by taking a ball, lifting its spherical boundary out of flat 3-space and squashing it to a point, enclosing a 4-dimensional interior. The stretched out ball innards are the 3-sphere, and this is what our brains are not conditioned to think of. $\endgroup$ Aug 16, 2021 at 5:05
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    $\begingroup$ @Edouard Maybe I should have phrased it differently, switch "think" with "visualize". $\endgroup$ Aug 16, 2021 at 10:43
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    $\begingroup$ A 3-sphere is curving in 4D space, analogously to how a 2-sphere is curving in 3D space. It is the 4D space part that we can't visualize, not the three dimensions of the 3-sphere's surface. We can think about four spatial dimensions via analogies and via formal mathematics but we can't think about four spatial dimensions "by intuition". $\endgroup$
    – jwezorek
    Aug 17, 2021 at 0:39
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When physicists and mathematicians say something is "a sphere", they mean a mathematical sphere. That could be a 3, 4, 5 or other dimensional sphere. So you have to ask "what is a sphere", outside normal language.

A sphere is defined as all the points in a space, that are the same distance from some fixed point. So in 3D its like the surface of a ball. In 2D we call it a circle (the "surface", or points equal distance from the centre, of a disc). In 4D and higher dimensions, we usually call it the surface af a hypersphere or N-sphere .

So when a cosmologist says the universe has a positive curvature and "should behave like a sphere", they don't necessarily mean "like a 3D sphere".

  • A universe that was shaped like an enormous 4D sphere, to our 3D senses, would look "flat", like any enormous sphere does. You have to ask what "flat" means for 4D space. A way of thinking about "flat" space is that if you measure the corners of trianges, they add up to 180 degrees, not more or less. Space as far as we can tell, seems almost perfectly flat. That tells us something about its structure.

  • A universe that was shaped like an enormous 4D sphere, to our 3D senses, would seem to be endless, but in fact would have finite size (even if we couldn't see it ourselves). If you traveled endlessly you'd potentially come back where you started. A lot of science fiction uses that idea. A problem with this idea is that we appear to be in a universe where space is expanding, so it may not be possible to travel round the universe, even if you had infinite time, because the "distance" youd have to travel would grow faster than you could travel it. Thats possible because the expansion of space itself isnt limited by the speed of light. Objects within space (including light and objects) cant travel faster than light. But when space expands, no objects move in space. Space itself is changing, and objects themselves aren't moving in space. That expansion is not limited by the speed of light. So the universe can become bigger, faster than you could travel round it. We believe this has actually happened. Thats why we talk about "the observable universe" - because we expect the rest of the universe we simply cannot and never will be able to see. Its too far away, moving too fast, light from it wont ever reach us.

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  • $\begingroup$ If I think of the surface of a potato then it seems to me possible, that a light beam in some direction on this surface would not come back to my position but might be traveling always without returning precisely to its beginning. This is just a guess, but to make this idea more resonable: might there be some "ball"-type object where the curvature has some irrational measure which would allow such no-return-lightbeams? (mm, perhaps this should be a separate question - but perhaps it is easy to answer just here ... ) $\endgroup$ Aug 16, 2021 at 10:40
  • $\begingroup$ Anything possible; we don't know enough to know which possible things are true. Nature has surprised us more than a few times. $\endgroup$
    – Stilez
    Aug 16, 2021 at 11:05
  • $\begingroup$ Upps, the latter seems to be misunderstanding,sorry: here I meant the mathematical / geometrical problem of the possible 3-d body with the above scribbled property... $\endgroup$ Aug 16, 2021 at 11:18
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    $\begingroup$ "In 2D we call it a disk, or circle." - no. A disk is the $2$ ball $D^2=B^2$ whilst a circle is the $1$-sphere $S^1$. A circle is the (one-dimensional) boundary of a disk. $\endgroup$
    – jacob1729
    Aug 16, 2021 at 14:13
  • $\begingroup$ @jacob1729 - fixed $\endgroup$
    – Stilez
    Aug 16, 2021 at 14:48
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You've run into a confusion, or ambiguity, that English shares with some (but probably not all) other languages. As the term is used in physics, a "sphere" does not actually have any center, as it is the surface of a "ball": The diagram which you've helpfully provided depicts a ball, whose volume does have a center. A sphere has a surface area, but not a volume, and even the description of its surface area is imprecise, because it must take account of that ratio (of the ball's circumference to its diameter) which is pi, and continues into an apparently endless series of variations.

However, even a ball only provides an instantaneous description of the figure (a possible shape of the universe) that you're concerned about, which is not a description complete enough to capture the figure as it's used in cosmology (your first tag), because the volume of the ball may be (and is, at least in that part of the cosmos which is observable by us) expanding overall: Consequently, time must be taken into account.

In Einstein's relativity, time is "visualized" as orthogonal to (in other words, "at right angles to") space: It can only be taken accurately into account through the use of at least two coordinate systems, which greatly complicates the appearance of the resulting figure, known as a "three sphere" or "glome". However, nothing physical like it can ever be seen from any known vantage point, because we do not actually see time: We only see evidence of our passage through it in objects, like the hands of clocks, that have at least some mass.

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    $\begingroup$ "Spherical" can refer either to a single sphere or to a no. of them, possibly nested inside each other: For instance, each of the "local universes" in Nikodem Poplawski's inflationary multiverse results from processes that occur on two different spatial scales, & is described, by him, as having a shape "like the skin of a basketball". $\endgroup$
    – Edouard
    Aug 15, 2021 at 18:47
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It like in a good video by nasa guy made some years ago. If there was 2D snails, moving on a platform, and their radius of vision is limited somewhat. Then a 3d creature can place endless pieces of same square, on every direction. As the snails go, they can't see back enough and not far enough (imagine this as a horizon). So they just move, and as big the canvas was made, as big the snail thinks it is. Even if 4 pieces would do the same illusion. On 3d creatures like us, it must be thought like we were on a balloon, balloon can be blown bigger and the objects go further. And round we go. In fourth dimension, it is like a hypercube, which has edges related to other edges of the cube. Like with snails, now the cube is filled with 3d blocks, and on and on we go with our rockets, but no edge will be there. It seems infinite, but it is just how 3d creature sees 4d space. Further explaining would need thinking, so this should be good :)

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