Cosmologist Mark Trodden has an excellent post where he disentangles three questions that most people are interested in knowing about the universe: What is its shape? Is it finite or infinite? Will it expand forever or eventually collapse back into itself?

It is sometimes mistakenly assumed that the three are variations on the same question and that answering one determines the answers to the other two. For example, if the universe if positively curved, then it must imply that it is also finite and will eventually recollapse. Trodden says that this is incorrect and explains what we need to know to answer each one, and concludes:

So the universe may be positively or negatively curved, or flat, and our solutions to GR tell us this. They may be finite or infinite, and connected up in interesting ways, but GR does not tell us why this is the case. And the universe may expand forever or recollapse, but this depends on detailed properties of the cosmic energy budget, and not just on geometry.

Well worth reading.

Marshall says

Hi Mano–biologist here, not a physicist, but I read this stuff casually. One question I have, that bugs me when people use the torus example. The article states, as I’ve read in many places, that homogeneity is a constraint–i.e. the curvature is the same (on a large scale, excluding gravity) everywhere in the universe. But, on a torus, the curvature of the side facing inwards necessarily has greater curvature than the outside, so homogeneity is violated. Ditto with the saddle (the middle picture of the NASA graphic)–in some locations, the curvature is concave, and at others it’s convex. Also, the saddle implies a form of symmetry around a central point, which would make that particular point special, which again more or less violates homogeneity.

My point is that the equations that describe these surfaces, that we use as examples, have special points, and the properties are location-dependent. Is this the case for the universe too? Wouldn’t this imply there’s a “center” of the universe?

mcrumiller says

Sorry, for the saddle, I meant that the curvature is direction-specific.

Mano Singham says

The curvature is specified at each

pointin space.The figures that are given to demonstrate the different types of curvature are extended out a bit so that we can visualize it, but the curvature is just the limiting case at the center of that figure. Connecting up adjacent points of the same curvature so as to get a surface that can be visualized can only be done for a few special cases, such as the flat curvature to give an infinite flat surface or a positive curvature that can be connected up to get a sphere.

With saddle curvature, we cannot draw a figure that connects up the different points. We only have the equations.

Does that help?

Marshall says

Perfectly, thank you!

Francisco Bacopa says

This is all beyond me. When we talk about space and curvature, aren’t we always talking about space without time? That’s because I know that from each of our perspectives, spacetime is most definitely positively curved.

Why? As I look out into space, I am not seeing things as they are, but as they were. If I look out in a flat circle at all things one light second away. I am seeing things as they were one second ago. The circumference of the circle will be tau times radius, or 2 x pi x radius for those who kick it old school.

But what if I have a powerful telescope and look out at the things a billion light years away? Hasn’t the universe expanded a lot during that billion years it has taken the light to reach me? So I am looking out into a

smalleruniverse, which is signified by all my observations being somewhat red shifted. Therefore, my circle will have a smaller circumference than tau x R (2pi x R).So spacetime as a whole is positively curved and closed on one end. But we already thought as the big bang started in a singularity. Is there another closed end, probably not according to my understanding.