The universe's size : always infinite?

[WannabeNewton]

Carroll's lecture notes don't mention anything about the 'finiteness' of the universe and the FAQ provides no reasoning behind why an open (or flat) universe is infinite.
I would have thought a flat universe could have come from a big bang, and therefore could be finite, yet expanding...

Well the flat space friedman model is generally assumed to be described by euclidean 3 - space which is infinite. It doesn't have to be euclidean, a 3 - torus is finite and closed and flat but it is generally assumed that k =0 corresponds to euclidean space.

 

[George Jones]

The Big Bang didn't happen at a point, it happened everywhere.

From the ffth paragraph od Carroll's notes:

We therefore consider our spacetime to be \mathbf{R} \times \Sigma, where \mathbf{R} represents the time direction and \Sigma is a homogeneous and isotropic three-manifold.

\Sigma is space. Equation (8.9) shows that for a flat universe, \Sigma is locally flat. Just below this equation, two possibilties for a locally flat \Sigma are given.

Globally, it could describe \mathbf{R}^3 or a more complicated manifold, such as the three-torus S^1 \times S^1 \times S^1

The first possibility above is for non-simply connected, infinite flat space that satisfies the cosmological principle, i.e., is homogeneous and isotropic The second possibility is for mutiply-connected, closed flat space that does not satisfy the cosmological principle (homogeneous but not isotropic).

I see that while I was typing, WannabeNewton posted about the same stuff.

 

[BruceW]

I think I get it now. The universe has always been infinite since the big bang, but in the early stages it was very dense but is now not as dense.

 

[George Jones]

I think I get it now. The universe has always been infinite since the big bang

Yes, this is true for a flat or open, homogeneous, isotropic universe.

but in the early stages it was very dense but is now not as dense.

Right! As t approaches zero, the density of the universe grows without bound everywhere.

 

[BruceW]

Understood. So when we say the universe began at a singularity, we mean the universe was always infinitely large, but it was a singularity everywhere, because it had arbitrarily great density?
And for a closed universe, it is as if 3-space is the surface of a sphere, which is why a closed universe always has been and always will be finite?
Our universe looks like its flat, but there is error on this, so it might actually be closed or open (it would only be slightly curved, but curved nonetheless). So this means we don't actually know whether the universe was always infinite (open) or whether it was once very small (closed). Is this right?
If there is always some error, will we ever know what type our universe is?

 

[bcrowell]

If there is always some error, will we ever know what type our universe is?

If our universe is really exactly flat, then every measurement of its curvature, no matter how precise, will always be statistically consistent with zero.

If our universe is not really exactly flat, then a sufficiently precise measurement would be statistically inconsistent with zero. However, it might take very advanced technology to achieve that precision.

 

[1mmorta1]

Your mathematical understanding of infinity is off. Infinity is not VERY LARGE, it is completely unending. There ARE, however, different sizes of infinity, measured in alephs. Take for example: The set of all integers. 0, 1,...infinity. This is an infinite set, but the set of all real numbers: 0.0000...1, 0.000...2, and so on, is also infinite. However, since there are just as many real numbers between 0 and 1 as there are integers, the infinite set of real numbers is larger than the infinite set of integers. As to the universe being infinite...I don't believe it is, and current models do not predict such a thing. I mostly just wanted to help you on your understanding of infinity :)

 

[Deuterium2H]

Your mathematical understanding of infinity is off. Infinity is not VERY LARGE, it is completely unending. There ARE, however, different sizes of infinity, measured in alephs. Take for example: The set of all integers. 0, 1,...infinity. This is an infinite set, but the set of all real numbers: 0.0000...1, 0.000...2, and so on, is also infinite. However, since there are just as many real numbers between 0 and 1 as there are integers, the infinite set of real numbers is larger than the infinite set of integers. As to the universe being infinite...I don't believe it is, and current models do not predict such a thing. I mostly just wanted to help you on your understanding of infinity :)

On the right track, but not quite technically right, 1mmortal1...

The reason the Reals constitute a larger Cardinality (higher infinity) then the Naturals is because it can be proved that A) The Set of Reals (R) is not bijective with the Naturals (N) , i.e. cannot be put in a one-to-one correspondence with N. and B), N is a Subset of the Reals.

You specific example is a bit misleading, because one could question why the Set of Prime Numbers is of the same size/Cardinality as N, even though one would intuitively think there must be far fewer Prime Numbers then there are Natural numbers.

 

[Jay Ssj]

ive read that universe is uses 70% of its energy in expanding itself. idk how it will help u but i just thought i shud post it :/


Btw m new and don't know much :P

 

[1mmorta1]

On the right track, but not quite technically right, 1mmortal1...

The reason the Reals constitute a larger Cardinality (higher infinity) then the Naturals is because it can be proved that A) The Set of Reals (R) is not bijective with the Naturals (N) , i.e. cannot be put in a one-to-one correspondence with N. and B), N is a Subset of the Reals.

You specific example is a bit misleading, because one could question why the Set of Prime Numbers is of the same size/Cardinality as N, even though one would intuitively think there must be far fewer Prime Numbers then there are Natural numbers.

You are correct, I should have brought up one - to - one correspondence, but I felt like it would just confuse people. The person who asked this question doesn't really understand the notion of infinity, so I thought I'd simplify things by painting a mental picture that may lead to some understanding of why there are different "sizes" of infinity without having to be too detailed.

 

[1mmorta1]

I will say, after reading fabric of the cosmos, I never got any feeling that the universe was infinite.

 

[BruceW]

I think bcrowell and george jones will disagree with you. (I personally don't know much about it, but they did seem to be saying that an open, homogeneous, isotropic universe is infinitely large).

Maybe immortal you are thinking of a closed universe? Which I think would be finite. The thing is that we currently aren't sure whether our universe is closed or open, so we don't know if it is finite or infinite.

 

[1mmorta1]

There are several good arguments for why our universe CANNOT be both infinite and homogeneous. The impression I get from most models is that our universe both finite and unbounded, so it could appear infinite from within. (Kind of like a sphere, there is no end or beginning. You can move in any direction on a sphere for an infinite amount of time and never reach a boundary, yet the amount of space on the surface of the sphere is finite)

 

[nazarbaz]

Aren't we confusing in this debate spatial concepts with notions about time ? Infinity and eternity ?
The expansion of the universe or the multiverse could be eternal, that does not imply the effective and actual infinity of space-time (or spaces-times).
Infinity is a virtuality, not a reality, then.

 

[BruceW]

There are several good arguments for why our universe CANNOT be both infinite and homogeneous. The impression I get from most models is that our universe both finite and unbounded, so it could appear infinite from within. (Kind of like a sphere, there is no end or beginning. You can move in any direction on a sphere for an infinite amount of time and never reach a boundary, yet the amount of space on the surface of the sphere is finite)

That model is the model of a closed universe. (Which our universe may or may not be).

In Stephen Hawkins' book, he mostly mentions the closed universe (which is finite but unbounded). Maybe that's where you got the impression from (that's where I got it from when I came on this thread a while ago).

 

[1mmorta1]

I actually never finished reading Hawkins' book, I thought it was terrible. I am familiar with the term closed(meaning finite but unbounded) from topology, but figured I would break it down a little. What I'm curious about is why everyone suddenly feels that the universe is infinite...I've never known this to be a popular belief. Even in Brian Greene's book, he doesn't seem biased to one position or the other, but describes different possible shapes for our universe with equal intensity.

 

[1mmorta1]

Also: The universe CAN be open and finite. It would just not be the unbounded universe we've been discussing...aka "There's a wall there."

 

[Jocko Homo]

Understood. So when we say the universe began at a singularity, we mean the universe was always infinitely large, but it was a singularity everywhere, because it had arbitrarily great density?
And for a closed universe, it is as if 3-space is the surface of a sphere, which is why a closed universe always has been and always will be finite?
Our universe looks like its flat, but there is error on this, so it might actually be closed or open (it would only be slightly curved, but curved nonetheless). So this means we don't actually know whether the universe was always infinite (open) or whether it was once very small (closed). Is this right?

As far as I know, this understanding is exactly right! However, I would add another interesting note...

If the universe were finite in spatial extent, the entire universe really would be "compressed" into a small region of space as you approach t = 0. Your revelation in this thread is that if the universe were infinite, it would remain infinite even as you approach t = 0 and it would "merely" be infinitely dense. However, I would argue that the two scenarios (finite or infinite universe) aren't very different. Even if the universe were infinite, everything in the universe will still have been "compressed" together in the sense that, for every two objects in this infinite universe, there will exist a point in time where those two objects are arbitrarily close together together... How weird is that?

If there is always some error, will we ever know what type our universe is?

In my humble opinion, in order to "know" that the universe is open, we'd need some new theory that is both accurate and would require an open universe. In other words, we'd need some new understanding of space and the universe...