# Does Big Bang Cosmology imply/require infinite space?

The reason I am asking this question is because if all points in space observe recession of galaxies the same as we do from Earth, the universe would have to be infinite (or a closed sphere in 4D or something. I know infinite space isn't a formal position of Big Bang cosmology, but is a non infinite space problematic in that worldview? Is an infinite cosmos thought the most likely? Basically I just want to know where this cosmology stands on infinity!

This is a good question because sometimes people argue there is evidence that the universe is infinite.

Basically when you assume space is isotropic and homogeneous then you are left with three options, a space that is flat and two other types. The two other types can easily be finite and I don't see much argument about it.

The flat one seems pretty easily to accommodate a finite space as well. It has a metric like $$d\tau^2=dt^2-(a(t))^2(dx^2+dy^2+dz^2).$$

And this is a purely local equation that could be applied to for instance the subset $$\{(t,\alpha,A,\beta,B,\gamma,C): \alpha^2+A^2=\beta^2+B^2=\gamma^2+C^2=1\}.$$

Where $dx$ measures your change in the $\alpha,A$ plane, $dy$ measures your change in the $\beta,B$ plane, and $dz$ measures your change in the $\gamma,C$ plane.

The solution isnt globally isotropic anymore even though locally it has a metric of the same form as the homogenous isotropic one. The form of the metric is a purely local construct and tells us almost nothing about the global topology.

This relates to you other question, in this situation everything looks the same locally in all directions as long as you look at coordinate distances less than 1. So as long as you can't see all the way around the universe then this solution will look the same in all directions at all locations, to the observer at that location.

Some people mistakeningly think that if your space is flat then your space has to be infinite. They are wrong, but would we think space is flat anyway?

Well when we measure that large scale curvature it could be flat, positive, or negative. So either it really is flat or else it is positive or negative but the value is so small that it happens to be smaller than the sensitivity of our equipment. We have good equipment so some people take that as evidence the universe is flat.

However there is a huge huge problem with that. Which is that we notice the past universe seems too uniform, so we think long ago they used to be close together which means the universe must have grown super fast for a while to get this big. But when the universe grows super fast a positive or a negative universe starts to look really really close to flat. Imagine a ball on that is huge or even the surface if the earth it is very flat.

So the universe looks very flat, but we have a theory that makes the universe become very close to flat even when they aren't exactly perfectly flat. So we don't know.

So we don't know if the universe is infinite. Is there evidence it is finite?

How would we know. You can look for repeats, like wandering around the earth and coming back to the same place you might wonder if things repeat if you go far enough.

But what if they did. A universe could have some regularity. If you were in the middle of a laser pulse you might think the universe is as big as the wavelength of the laser because when you go that distance everything looks the same, but if you went far enough you'd finally get to where the laser started or stopped and things would start to look different.

So no matter how many times you travelled to a placed that looked like the place where you started how would you know if the universe is finite or just large with really regular repetitions.

So we aren't going to know either way. There is no possible data, nothing, that forces the universe to be infinite (it could always be finite and just looks a certain way near us) and there is no possible data, nothing, that forces the universe to be finite (it could always be infinite and have really regular repetitions near us).

If you read that last paragraph you see it isn't really a scientific question anything less than a perfect observation (which is not possible) gives enough room to have both possibilities. So it isn't a scientific question.

We don't know. We can't know.

• It's easy to create spacetimes with holes cut out of them, etc., as you've done here. It's a pointless exercise because we have no reason to think spacetime has structure like this, and if structure like this did exist, our theories would not have any useful way to make predictions about those structures. For these reasons, we usually are interested in the maximal extension of any spacetime. – Ben Crowell May 8 at 5:35