# The universe's size : always infinite?

• pedersean
In summary: So, if the universe is infinite it would have to be expanding infinitely fast, as there is an infinite amount of space to contribute to the expansion. But when you take an infinitely small portion of the universe (like what we can measure) that expansion rate can be finite. So if the universe were to gain 10% size over a given time, it could do so and still be infinite as 1.1*∞ = ∞.
pedersean
I came across a startling position on more than one occasion while reading "The Fabric of the Cosmos: Space, Time, and the Texture of Reality" by Brian Greene. The position is that our immeasurable universe is infinite. He continues by writing that any mathematical modification to the size of the universe will always result infinite. Perhaps my understanding of infinity is misleading, but I've always held the position of infinity being obtainably intangible and impossible. Instead I prefer to hold that infinity is instead a special numerical placeholder of the extraordinarily large, and with subtracting from infinity a given tolerance becomes greater. Adding to infinity decreases that tolerance by bringing the actual number closer to its equivalent of the all powerful forever number. Am I nuts to argue that an expanding universe can not persist its size?

It basically which side of the fence you are on

If you think that the universe was created by the big bang, then it is physically impossible to fit infinite mass inside a ball the size of a pea (yes I know, laws of physics has been broken before). Then I would say that the universe is expading an it does have it's limits if you consider the theory that energy cannot be created or destroyed.

If you are one of those people who say that the universe has been here since the dawn of time and will always be here, then no... the universe isn't expanding (somehow :P)

as for the question is it infinite, I believe that rather than the universe being never ending, I think its in a big loop (like how people thought that Earth was flat, but it's in a sphere, if you go in one direction for long enough you will eventually arrive in the exact same place that you have started from. ;)

Live Long and Prosper \ m /

Infinity is a funny thing. For example, let's say you have an infinite amount of water. That means for every 1 oxygen atom you have 2 hydrogen atoms. While you obviosly have 2x as many hydrogen atoms, you at the same time have equal amounts, since there are infinite amounts of both hydrogen and oxygen. (1*∞)/3 = (2*∞)/3

If you take 1/2 of an infinite volume you still end up with an infinite volume. ∞/2 = ∞. Only if you divide an infinite volume infinitely do you get a finite number: (2*∞)/∞ = 2

So, if the universe is infinite it would have to be expanding infinitely fast, as there is an infinite amount of space to contribute to the expansion. But when you take an infinitely small portion of the universe (like what we can measure) that expansion rate can be finite. So if the universe were to gain 10% size over a given time, it could do so and still be infinite as 1.1*∞ = ∞.

All this seems counter intuitive and hard to accept, but the math is the math. Without accepting infinity as a real possibility than things like a singularity become impossible. An infinite universe also nicely explains expansion. While the universe was a finite size each point had an infinite amount of energy contributing to its expansion. Only once it reached an infinite size would there be a finite amount of energy at each point and be able to start to cool down.

Hope this helps.

The current theory of the universe says that the universe is of finite size.
But there are models of the universe that say it could be infinite.
At the moment, they think its finite.

Wouldn't an infinite universe imply infinite energy? For example, two objects infinitely away from each other would have infinite potential energy. What if 2 objects fall at each other from opposite "sides" of an infinite universe, would they accelerate more and more as they got closer to each other? Would they be infinitesimally approaching the speed of light as they accelerate closer to each other, or would it just take infinite time for this to ever happen anyway?

Well for all intents and purposes the universe is infinite, in the fact that if we travel from one end to the other we find ourselves back at the place from which we started.

WMAP has found, to some degree of experimental error, that the universe is flat. This means that when one looks at the Friedmann model for a flat universe it is infinite in extent not finite. However, we can only view a finite portion (observable universe) of it because there are regions of the universe that are expanding faster than the speed of signals from those areas.

DragonPetter said:
Wouldn't an infinite universe imply infinite energy? For example, two objects infinitely away from each other would have infinite potential energy. What if 2 objects fall at each other from opposite "sides" of an infinite universe, would they accelerate more and more as they got closer to each other? Would they be infinitesimally approaching the speed of light as they accelerate closer to each other, or would it just take infinite time for this to ever happen anyway?

Yes an infinite universe implies infinite energy. If the universe was empty execpt for those two obejects then yes they theoretically would fall towards at speeds approaching the speed of light, and yes they would never actually reach each other. It is also possible that gravitons from one object would never reach the other and the falling would never even begin. We don't know enough about gravity to say for sure.

sicarius said:
Infinity is a funny thing. For example, let's say you have an infinite amount of water. That means for every 1 oxygen atom you have 2 hydrogen atoms. While you obviosly have 2x as many hydrogen atoms, you at the same time have equal amounts, since there are infinite amounts of both hydrogen and oxygen. (1*∞)/3 = (2*∞)/3

If you take 1/2 of an infinite volume you still end up with an infinite volume. ∞/2 = ∞. Only if you divide an infinite volume infinitely do you get a finite number: (2*∞)/∞ = 2

So, if the universe is infinite it would have to be expanding infinitely fast, as there is an infinite amount of space to contribute to the expansion. But when you take an infinitely small portion of the universe (like what we can measure) that expansion rate can be finite. So if the universe were to gain 10% size over a given time, it could do so and still be infinite as 1.1*∞ = ∞.

All this seems counter intuitive and hard to accept, but the math is the math. Without accepting infinity as a real possibility than things like a singularity become impossible. An infinite universe also nicely explains expansion. While the universe was a finite size each point had an infinite amount of energy contributing to its expansion. Only once it reached an infinite size would there be a finite amount of energy at each point and be able to start to cool down.

Hope this helps.

Wow, that was interesting.

But can there be an infinite velocity? I mean, Einstein proved that the speed of light can not be breached. Unless you are telling me that space needs not follow that rule.

If so it would be pose very interesting questions to space and its affects.

Nano-Passion said:
Wow, that was interesting.

But can there be an infinite velocity? I mean, Einstein proved that the speed of light can not be breached. Unless you are telling me that space needs not follow that rule.

If so it would be pose very interesting questions to space and its affects.

Precisely, space does not follow that rule.

WannabeNewton said:
Precisely, space does not follow that rule.

Space is indeed interesting and mysterious. It really challenges your imagination and reasoning to the extreme. 0__o

Too many people take the word "space" for granted. We all grew up in it. Space to most people is just the room in their kitchen. Most people think of space as nothing. But in physics space actually has a life and physics of its own. Wow, I only got to appreciate it when I started to deeply ponder..

I wonder how much we truly understand about it?

Nano-Passion said:
Wow, that was interesting.

But can there be an infinite velocity? I mean, Einstein proved that the speed of light can not be breached. Unless you are telling me that space needs not follow that rule.

If so it would be pose very interesting questions to space and its affects.

Thank you.

The speed of light is a measure of movement through space, and that speed cannot be breached. Expanding space is not "moving through space" and does not hit the same limitations.

sicarius said:
Thank you.

The speed of light is a measure of movement through space, and that speed cannot be breached. Expanding space is not "moving through space" and does not hit the same limitations.

Yes I've heard that before -- though I wonder about the mathematics behind these sort of things and I'm dumbfounded how we can use math to describe phenomena such as that. Unless there is no rigorous math to it and it is philosophical reasoning to the idea of expanding space.

Nano-Passion said:
Yes I've heard that before -- though I wonder about the mathematics behind these sort of things and I'm dumbfounded how we can use math to describe phenomena such as that. Unless there is no rigorous math to it and it is philosophical reasoning to the idea of expanding space.

I may be wrong here, but I think that it is more like they have not found any math that disallows this, not so much as they have mathematically proved it.

There is a rigorous mathematical explanation for the universe being able to expand faster than the speed of light. It is called general relativity.
The universe is highly curved at large scale, which is why two faraway objects can be moving away from each other faster than the speed of light.

BruceW said:
There is a rigorous mathematical explanation for the universe being able to expand faster than the speed of light. It is called general relativity.
The universe is highly curved at large scale, which is why two faraway objects can be moving away from each other faster than the speed of light.

The universe is flat, according to observations, at the large scale and two objects don't really move faster than each at the speed of light but rather the space between them expands faster than the speed of light.

Closed, pending moderation.

Zz.

Hi, pedersean,

Welcome to PF!

It would have been better to post this in Cosmology rather than in General Physics. Typically people who are most knowledgeable about a particular field will only pay attention to posts in the relevant forum. Because the discussion had drifted off track, the thread was temporarily locked. I've moved it to Cosmology and opened it back up again.

pedersean said:
I came across a startling position on more than one occasion while reading "The Fabric of the Cosmos: Space, Time, and the Texture of Reality" by Brian Greene. The position is that our immeasurable universe is infinite.
This is not really right, and since Brian Greene is a competent physicist, I think probably what's happened is that you misinterpreted or oversimplified something he wrote. We have an entry on this topic in the cosmology FAQ: https://www.physicsforums.com/showthread.php?t=506986 We actually don't know whether the universe is spatially finite or spatially infinite.
pedersean said:
He continues by writing that any mathematical modification to the size of the universe will always result infinite.
This sounds like another case where the message got garbled somewhere along the line. This would depend on what was meant by "mathematical modification."

pedersean said:
Perhaps my understanding of infinity is misleading, but I've always held the position of infinity being obtainably intangible and impossible.
The Math FAQ has a good entry on infinity:
https://www.physicsforums.com/showthread.php?t=507003 The truth or falsehood of your statement would depend on what you meant by "obtainably intangible and impossible."

pedersean said:
Instead I prefer to hold that infinity is instead a special numerical placeholder of the extraordinarily large, and with subtracting from infinity a given tolerance becomes greater. Adding to infinity decreases that tolerance by bringing the actual number closer to its equivalent of the all powerful forever number.
This kind of statement really can't be decided, because it uses undefined terms like "all powerful forever number." The real number system doesn't include infinite numbers. The math FAQ entry gives some examples of number systems that do include infinite numbers.

pedersean said:
Am I nuts to argue that an expanding universe can not persist its size?
Not nuts, just incorrect :-) I'm not clear here on why you use the word "persist." Are you discussing the possibility that it would start out infinite and then become finite at some later time? (This would seem to go along with what you said above about "mathematical modification.") According to general relativity, if the universe is finite at one time, then it's finite at all earlier and later times; if it's infinite at one time, then it's infinite at all earlier and later times. This can be proved mathematically based on the Einstein field equations plus some other very reasonable physical assumptions that we have good reason to believe hold in our universe: http://arxiv.org/abs/gr-qc/9406053 The term for this is "topology change."
Then I would say that the universe is expading an it does have it's limits if you consider the theory that energy cannot be created or destroyed.
This actually doesn't quite work in cosmology. There is no principle of conservation of energy in cosmology. We have a FAQ entry about this: https://www.physicsforums.com/showthread.php?t=506985

If you think that the universe was created by the big bang, then it is physically impossible to fit infinite mass inside a ball the size of a pea
Mass and energy are equivalent in relativity, so we actually can't define the total mass of the universe (regardless of whether it's spatially finite or spatially infinite). However, we can discuss things like how many hydrogen atoms there are. "The size of a pea" would only apply to cosmologies that are spatially finite (and therefore spatially finite at all times). In these cosmologies, there is only a finite number of hydrogen atoms (or any other particle) in the universe.

sicarius said:
Infinity is a funny thing. For example, let's say you have an infinite amount of water. That means for every 1 oxygen atom you have 2 hydrogen atoms. While you obviosly have 2x as many hydrogen atoms, you at the same time have equal amounts, since there are infinite amounts of both hydrogen and oxygen. (1*∞)/3 = (2*∞)/3

If you take 1/2 of an infinite volume you still end up with an infinite volume. ∞/2 = ∞. Only if you divide an infinite volume infinitely do you get a finite number: (2*∞)/∞ = 2
The thing to be careful about here is that unless you specify a particular number system (with certain axioms), these statements about arithmetic operations involving infinity are neither true not false. You also have to be careful about your implicit assumption that there is only one infinite number, which is not true in all number systems that include infinite numbers. The math FAQ entry does a good job of explaining this.

sicarius said:
So, if the universe is infinite it would have to be expanding infinitely fast, as there is an infinite amount of space to contribute to the expansion. But when you take an infinitely small portion of the universe (like what we can measure) that expansion rate can be finite. So if the universe were to gain 10% size over a given time, it could do so and still be infinite as 1.1*∞ = ∞.
This is sort of right, except that you haven't really defined what you meant by "infinitely fast." Maybe you mean the velocity of one galaxy relative to another galaxy that is at a cosmological distance from it? In this case, there is actually no uniquely defined way to talk about the velocity in GR. However, one reasonable way to talk about it is to let $v=\Delta L/\Delta t$, where L and t are the quantities defined in this cosmology FAQ entry: https://www.physicsforums.com/showthread.php?t=506990 In that case, v is finite for any two galaxies, but in an infinite universe there is no upper bound on v (and v can be greater than c).

sicarius said:
While the universe was a finite size each point had an infinite amount of energy contributing to its expansion. Only once it reached an infinite size would there be a finite amount of energy at each point and be able to start to cool down.
This is incorrect, because, as discussed above, GR says changes of topology aren't possible.

BruceW said:
The current theory of the universe says that the universe is of finite size.
But there are models of the universe that say it could be infinite.
At the moment, they think its finite.
Nope. The cosmology FAQ entry discusses this.

Haroldingo said:
Well for all intents and purposes the universe is infinite, in the fact that if we travel from one end to the other we find ourselves back at the place from which we started.
No, the wrap-around thing would apply to a spatially finite universe (one with finite volume), but as explained in the FAQ, we don't know if it's spatially finite or spatially infinite.

WannabeNewton said:
WMAP has found, to some degree of experimental error, that the universe is flat. This means that when one looks at the Friedmann model for a flat universe it is infinite in extent not finite. However, we can only view a finite portion (observable universe) of it because there are regions of the universe that are expanding faster than the speed of signals from those areas.
This is not quite right. As explained in the FAQ entry, the universe is within error bars of being flat. Therefore it could have either positive curvature (with finite spatial volume) or negative curvature (with infinite spatial volume).

sicarius said:
Yes an infinite universe implies infinite energy. If the universe was empty execpt for those two obejects then yes they theoretically would fall towards at speeds approaching the speed of light, and yes they would never actually reach each other. It is also possible that gravitons from one object would never reach the other and the falling would never even begin. We don't know enough about gravity to say for sure.
Sorry, but this is basically all wrong.

Nano-Passion said:
But can there be an infinite velocity? I mean, Einstein proved that the speed of light can not be breached. Unless you are telling me that space needs not follow that rule.
It's not an question of space versus physical objects, it's a question of local versus global. Relativity only prohibits objects from zooming right past each other at >c. For cosmologically distant objects, velocity isn't even uniquely well defined (see above).

Nano-Passion said:
Yes I've heard that before -- though I wonder about the mathematics behind these sort of things and I'm dumbfounded how we can use math to describe phenomena such as that. Unless there is no rigorous math to it and it is philosophical reasoning to the idea of expanding space.
It is rigorous math. It's how general relativity works.

WannabeNewton said:
two objects don't really move faster than each at the speed of light but rather the space between them expands faster than the speed of light.
This is a common way of explaining it nonmathematically. Mathematically, "speed" is just not defined in this context, and expansion of space, although a possible verbal description, is not the only way of verbally describing the mathematics of an expanding universe.

-Ben

Last edited by a moderator:
Originally Posted by DragonPetter
Wouldn't an infinite universe imply infinite energy?

No, not if the universe is a mixture of positive and negative energy, unevenly distributed on small scales. Then you could have an infinite universe with zero or really any finite amount of energy.

Bob3141592 said:
No, not if the universe is a mixture of positive and negative energy, unevenly distributed on small scales. Then you could have an infinite universe with zero or really any finite amount of energy.

bcrowell said:
This is a common way of explaining it nonmathematically. Mathematically, "speed" is just not defined in this context, and expansion of space, although a possible verbal description, is not the only way of verbally describing the mathematics of an expanding universe.

-Ben

The only thing I can think of is that the scale factor $$\dot{a} > 0$$ and $$\ddot{a} < 0$$. What other way is there if I may ask? I have only gone as far as Wald in terms of texts so this is the only way I have learned to see it ><

WannabeNewton said:
The only thing I can think of is that the scale factor $$\dot{a} > 0$$ and $$\ddot{a} < 0$$. What other way is there if I may ask? I have only gone as far as Wald in terms of texts so this is the only way I have learned to see it ><

Well, those are mathematical ways of describing it. Verbally, you can say that (a) galaxies A and B are moving away from each other, or (b) galaxies A and B are both at rest, but space between them is expanding. Actually, if you like, you can say that (c) galaxies A and B are moving toward each other, but the space between them is expanding so fast that it more than cancels out the effect of their motion.

bcrowell said:
Well, those are mathematical ways of describing it. Verbally, you can say that (a) galaxies A and B are moving away from each other, or (b) galaxies A and B are both at rest, but space between them is expanding. Actually, if you like, you can say that (c) galaxies A and B are moving toward each other, but the space between them is expanding so fast that it more than cancels out the effect of their motion.

Oh I see what you mean. Thanks mate.

For us the universe is infinite. Albert Einstein said that nothing can exceed the speed of light (186,000 miles per second). However, this law does not apply to the expansion of our universe. This law only applies when an object is moving from one place to another withing our universe. However, the universe can expand at speeds exceeding the speed of light. So basically, if the universe is expanding at a speed greater then the speed of light, we will never be able to reach the edge of the universe, even if we travel at 99.999% of the speed of light. So, this makes the universe infinite for us. However if the universe is not expanding at a speed greater then the speed of light, then I'm not sure if the universe should be considered infinite or not.

mdmaaz said:
So basically, if the universe is expanding at a speed greater then the speed of light, we will never be able to reach the edge of the universe, even if we travel at 99.999% of the speed of light. So, this makes the universe infinite for us.

Umm.. now I'm confused. I was under the impression that the universe is considered infinite full-stop, not qualified by "for us".

narrator said:
Umm.. now I'm confused. I was under the impression that the universe is considered infinite full-stop, not qualified by "for us".

I think the poster is referring to ideas of causally connected space/time. Ie: the OU is big enough and expanding fast enough that it may as well be infinite as we are unlikely to see/influnce anything beyond our particle horizon.

bcrowell - are you saying the universe is 'infinite' simply because it keeps expanding? Because when the universe was at small cosmological time, surely it was pretty small (i.e. finite)?

BruceW said:
bcrowell - are you saying the universe is 'infinite' simply because it keeps expanding?

No. Have you read the FAQ? https://www.physicsforums.com/showthread.php?t=506986 If there's something specific in it you're having trouble understanding, please post a question about that.

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...

BruceW said:
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.

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.

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.

BruceW said:
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.
BruceW said:
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.

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?

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