Can space only be infinite?

Question

I have read before that if you could just go fast enough, as a thought experiment, and you move in a straight line, in any direction, that you eventually might reach the spot from which you started. I mean, that would happen if the universe had such a "shape". If space would be curved in such a way, if the universe would be akin to a 4 dimensional sphere.

But how could such a thing even be possible? I mean, if we would imagine a 2 dimensional flatland universe, with a positive curvature of space, so that if you move in any direction, by moving fast enough, you would end up in the same spot eventually, that would be a bit like what we can do on Earth, moving in any direction along the surface and ending up in the same spot eventually. But we can do this because we have 3 dimensions in which the Earth is in. I mean, this flatland universe, would it be "in" anything? And I guess the answer would be no, because it's the universe and nothing is "outside" it. But isn't nothing.. something? I mean, nothing as the total absence of space, time, energy and matter, can it exist? I know, that's a second question, but it goes along with the first one. I don't think it's a stupid question, if a universe is "inside" a total absence of anything, doesn't this total absence of anything exist? I mean this universe wouldn't really be inside anything because nothing, isn't anything. But still.. wouldn't this nothing somehow exist as nothing? It's mindboggling to me, for sure, but this may all go away if space is infinite and flat and I understood this may be the case, from calculations based on the background radiation. In fact, if space is 3 dimensional and infinite... how could we even know if there isn't anything far far away, like a quintillion lightyears away? Something that didn't originate from our universe.. but from a seperate big bang? Can't space be independent of the big bang, as something that existed before it? Just perhaps? Or not? And if not, wouldn't physicists have to explain how space came into existence?

Answer 1

"But we can do this because we have 3 dimensions in which the Earth is in." - And that's where you're wrong: We generally simply consider our universe - our spacetime - to be some sort of manifold, but we do not by default embed this manifold in anything: this geometric shape just exists on its own, not as a subset.

That a sphere (like the Earth approximately is) has the property that you can return to your starting point without ever changing direction is not a consequence of any dimensions it's embeddded in - you can show that purely in the two-dimensional description of the sphere itself.

See also this question and its answers for discussions about how one would go about embedding spacetime into higher dimensions - notably, for arbitrary four-dimensional spacetimes you would need a 252-dimensional ambient space in the "worst" case.

See this question and its answers for the practical question of whether our actual universe is finite or infinite.

Answer 2

It depends how you interpret the word 'space' in your question. If you understand it as a mathematical space, then certainly it can be finite. You can interpret the surface of sphere for instance as a 2-dimensional finite space.

However, a surface in general is merely a mathematical abstraction. It does not exist in reality. The 'surface' of any object in our reality is only a psychological impression given by a certain regularity of the arrangements of the atoms making up the object. If you examine it under a strong microscope you can see that there is no surface. All there is is individual atoms arranged in a certain way in our 3-D space. And you can actually go from one point on the 'surface' of the object to another in a straight line just by drilling through the object.

If you are asking instead whether our physical 3D-space can be finite, this could obviously only be answered by proving it directly e.g. by showing that eventually a straight line returns into itself. Conceptually, this would be at least questionable however, as it is based on mathematical abstractions, which, as explained above, do actually not have a counterpart in physical reality.