Why Is a Flat Universe Infinite?

[albie]

I cannot find an answer online. Help please.

 

[Halc]

Why Is a Flat Universe Infinite?

If it is unbounded (has no edge), then flat space is infinite for the same reason a flat Euclidean 2d surface (flat unbounded paper say) must be infinite. If it is positively curved (like the surface of Earth), then it can be finite despite the lack of an edge.

 

[Ibix]

Because our models of the universe are everywhere homogeneous and isotropic. Something with an edge isn't homogeneous and the only way to get a flat universe with finite volume is to wrap it into something like a torus, and that isn't isotropic.

 

[jbriggs444]

Because our models of the universe are everywhere homogeneous and isotropic. Something with an edge isn't homogeneous and the only way to get a flat universe with finite volume is to wrap it into something like a torus, and that isn't isotropic.

One should be able to wrap around the faces of a three-cube so that the left face is identified with the right face, the top face with the bottom face and the front face with the back face. This is the three dimensional analogue of the screen in Asteroids or Pac Man.

This could be finite, flat and isotropic

A torus embedded in three space and using the induced metric is indeed not isotropic. But there is no requirement that the space in question be embedded in a higher dimensional space or that it use the induced metric.

Or am I missing something? Never having taken a course in topology, such is a possibility...

Oops. I am missing something. If you lay out a grid of landmarks in the wrapped-around three-cube, there are directions where you can lay down more landmarks before seeing a repetition. The space has a detectable directionality. It may be homogeneous on a large scale, but it is not isotropic if you look carefully. Or, looking at it another way, there are countably many angles at which you can look down grid lines with rational slopes and see yourself. Distributed in a way that makes the grid obvious if you can measure distance. Plus uncountably many angles with irrational slopes along which the sight line never intersects any of your wrapped around positions.

 

[timmdeeg]

Because our models of the universe are everywhere homogeneous and isotropic. Something with an edge isn't homogeneous and the only way to get a flat universe with finite volume is to wrap it into something like a torus, and that isn't isotropic.

But why should a universe in case it is anisoptropc and the cosmological principle doesn't hold but which is still in the frame of the EFE have an "edge"?

 

[Ibix]

But why should a universe in case it is anisoptropc and the cosmological principle doesn't hold but which is still in the frame of the EFE have an "edge"?

I'm not sure what you mean. I was saying a homogeneous and isotropic universe can't have an edge if it's flat. But non-homogeneity doesn't imply an edge - for example Kerr spacetime is non-homogeneous and doesn't have an edge.

 

[timmdeeg]

I'm not sure what you mean. I was saying a homogeneous and isotropic universe can't have an edge if it's flat. But non-homogeneity doesn't imply an edge - for example Kerr spacetime is non-homogeneous and doesn't have an edge.

Sorry, I misunderstood you.

 

[albie]

Because our models of the universe are everywhere homogeneous and isotropic. Something with an edge isn't homogeneous and the only way to get a flat universe with finite volume is to wrap it into something like a torus, and that isn't isotropic.

Sorry but doesn't make sense to me. Are you saying that an infinite universe has planets and stars going on for ever or are you saying that scientists are counting the void beyond the universe as part of our universe?

You use words like 'homogenous' and 'isotropic' but that implies a measurement. How can you measure infinity?

 

[Ibix]

Are you saying that an infinite universe has planets and stars going on for ever

Yes. That's what "homogeneous" means - the same everywhere.

or are you saying that scientists are counting the void beyond the universe as part of our universe?

There is no such thing that we're aware of.

You use words like 'homogenous' and 'isotropic' but that implies a measurement.

No, it implies a judgement that what we can see and measure of the universe means that modelling it as homogeneous is sensible. It clearly isn't homogeneous on the small scale (punch the air then punch a table and see if it feels the same), but if we look out further and further on larger scales the more same-y everything looks. Thus we make an assumption that on really large scales everything is more or less the same, feed that in to our maths, and see what it says. What it says turns out to include prediction of the cosmic microwave background and cosmological redshifts (subsequently confirmed by observation), and that there are three classes of possible universe of this type, two of which are infinite.

How can you measure infinity?

We can't. What we can do is extrapolate based on what we can see of the universe, and that extrapolation says the universe is either infinite, or a closed 3-sphere that is very very large.

 

[Jaime Rudas]

Because our models of the universe are everywhere homogeneous and isotropic. Something with an edge isn't homogeneous and the only way to get a flat universe with finite volume is to wrap it into something like a torus, and that isn't isotropic.

Well, but, that our models of the universe are homogeneous and isotropic does not necessarily mean that the real universe is also. It could be that the real universe is a finite flat torus of a volume several orders larger than the observable universe.

 

[PeroK]

You use words like 'homogenous' and 'isotropic' but that implies a measurement.

There is no such logical implication. Homogeneous and isotropic can and do apply to a (mathematical) model of our universe. These properties are used to make predictions of what can be measured. The measurements, as in all physics, in general either corroborate the model or not.

How can you measure infinity?

The infinite extent of our universe may always be something of a conjecture. Note that not everyone agrees with this. For example, we know with certainty that the Earth is finite. It's not clear what experiment could confirm with certainty that our universe is infinite.

 

[Ibix]

Well, but, that our models of the universe are homogeneous and isotropic does not necessarily mean that the real universe is also. It could be that the real universe is a finite flat torus of a volume several orders larger than the observable universe.

Of course. But it seems odd that the universe is locally isotropic but globally anisotropic - why wouldn't whatever process picked out the special directions in the topology not affect mass distribution and geometry? It's not impossible, no, but it's adding complexity to the model in a way that is deliberately undetectable and with no theoretical justification beyond "it could be true".

So yes, the universe could be more complicated than our extrapolation because you can always invoke additional complexity with the only constraint being that it can't affect the model's predictions for existing measurements. But in that case why stop at a torus? We could have a universe shaped like a taurus, as long as the scale is sufficiently large.

 

[PeterDonis]

It could be that the real universe is a finite flat torus of a volume several orders larger than the observable universe.

But in the absence of any evidence for this, your proposed model here gets scraped right off by Occam's Razor.

 

[Vanadium 50]

The universe has no boundary or edge.
A 2D flat surface with no boundary or edge must be infinite.

We use "flat" to mean a 3D analogy to the 2D term. This is one of the properties carried over by analogy. Put another way, if it did not have this property, we'd probably call it something else.

 

[PeterDonis]

A 2D flat surface with no boundary or edge must be infinite.

A flat 2-torus is a counterexample to this. Such a manifold can't be embedded in Euclidean 3-space (as an ordinary curved 2-torus can), but that doesn't mean it doesn't exist.

 

[Jaime Rudas]

It's not impossible, no, but it's adding complexity to the model in a way that is deliberately undetectable and with no theoretical justification beyond "it could be true".

Yes, that's right, there is currently no theoretical justification for it, but there might be in the future. For example, if in the future it were discovered that the total energy can't be infinite.

 

[PeterDonis]

there is currently no theoretical justification for it

There is currently no observational justification for it. It's perfectly possible to construct theoretical models with the property you describe. We just don't see any evidence for them.

if in the future it were discovered that the total energy can't be infinite

Any such "discovery" would have to be experimental.

 

[albie]

Yes. That's what "homogeneous" means - the same everywhere.

There is no such thing that we're aware of.

No, it implies a judgement that what we can see and measure of the universe means that modelling it as homogeneous is sensible. It clearly isn't homogeneous on the small scale (punch the air then punch a table and see if it feels the same), but if we look out further and further on larger scales the more same-y everything looks. Thus we make an assumption that on really large scales everything is more or less the same, feed that in to our maths, and see what it says. What it says turns out to include prediction of the cosmic microwave background and cosmological redshifts (subsequently confirmed by observation), and that there are three classes of possible universe of this type, two of which are infinite.

We can't. What we can do is extrapolate based on what we can see of the universe, and that extrapolation says the universe is either infinite, or a closed 3-sphere that is very very large.

Yes. That's what "homogeneous" means - the same everywhere.

There is no such thing that we're aware of.

No, it implies a judgement that what we can see and measure of the universe means that modelling it as homogeneous is sensible. It clearly isn't homogeneous on the small scale (punch the air then punch a table and see if it feels the same), but if we look out further and further on larger scales the more same-y everything looks. Thus we make an assumption that on really large scales everything is more or less the same, feed that in to our maths, and see what it says. What it says turns out to include prediction of the cosmic microwave background and cosmological redshifts (subsequently confirmed by observation), and that there are three classes of possible universe of this type, two of which are infinite.

We can't. What we can do is extrapolate based on what we can see of the universe, and that extrapolation says the universe is either infinite, or a closed 3-sphere that is very very large.

>>

No, it implies a judgement that what we can see and measure of the universe means that modelling it as homogeneous is sensible.

I asked if it was a measurement and then you said no but then used the word measurement in the same sentence. Just saying.

>>or a closed 3-sphere that is very very large.
So it isn't certain that the universe is infinite and flat. I will research a closed 3-sphere. Thanks for that.

Also, how do we gauge that the universe is not curved? Wouldn't a curved universe appear to be flat even to instruments? If we are searching for microwaves wouldn't we see an infinite loop if we went looking for them? if the universe was curved, I mean.

 

[Ibix]

I asked if it was a measurement and then you said no but then used the word measurement in the same sentence. Just saying.

I said the infinite size was an extrapolation from measurements, not a measurement itself.

So it isn't certain that the universe is infinite and flat. I will research a closed 3-sphere. Thanks for that.

Infinite and flat is our current best fit model, but infinite and negative curvature or finite and closed (but very large) are both possibilities.

Also, how do we gauge that the universe is not curved?

You can measure the scale of the fluctuations in the cosmic microwave background. How they look depends slightly on the curvature of spacetime.

 

[Ibix]

I asked if it was a measurement and then you said no but then used the word measurement in the same sentence. Just saying.

I said the infinite size was an extrapolation from measurements, not a measurement itself.

Note that this is actually quite a general phenomenon. We observe or measure something, which leads us to propose something we can't actually directly measure, which lets us make new predictions.

For example we notice that when we let go of things they fall. From careful numerical study we eventually realise that there are patterns to the way things fall and propose a force of gravity. From that we can predict things like the location of planets we haven't seen yet. But nobody has ever seen "the force of gravity". You can't go to a museum or a zoo or something and see the force of gravity - you can only see that when we drop things they fall. The force of gravity is a model that is based on numerical analysis of observational data and can be used to make predictions.

Similarly we have observational data on our region of the universe, and it looks reasonably homogeneous and isotropic. So we try assuming that the whole universe is homogeneous and isotropic. Given our model of gravity, this assumption implies that the universe is either negatively curved and infinite, flat and infinite, or positively curved and closed, and is expanding (or if it is closed, it may be contracting). Further measurements tell us that the universe is either flat or very nearly so and expanding.

So "the universe is spatially flat and infinite" is a true statement about one model of the universe. It's the model that best fits the observations, but it is not directly an observation or measurement itself. It would be foolish to discard the model's claims about the rest of the universe out of hand, because it rests on fairly solid foundations, but it is a model and we can only measure things about the part of the universe we can see.

 

[vanhees71]

To be honest, one should mention the "Hubble tension", i.e., the difference in the determination of the Hubble constant when measured via "early-time probes" (like the fluctuations of the CMBR and based on the assumption of the concordance $\Lambda\text{CDM}$ standard model of cosmology) and "late-time probes" (model independent determinations of the distance-redshift relation). Despite the problems with the latter "model-independent determinations" due to trouble with dust, etc., the discrepancy seems to consolidate recently also using other observables. Despite possible systematic quibbles one explanation could be that we live in a less dense region than on average. Here's a pretty recent review:

https://doi.org/10.1088/1361-6382/ac086d (open access)

 

[Jaime Rudas]

Despite the problems with the latter "model-independent determinations" due to trouble with dust, etc., the discrepancy seems to consolidate recently also using other observables. Despite possible systematic quibbles one explanation could be that we live in a less dense region than on average. Here's a pretty recent review:

https://doi.org/10.1088/1361-6382/ac086d (open access)

It's not clear to me what this has to do with the topic of the thread, but a more recent review on the issue of tension in $H_0$ can be seen here:

https://arxiv.org/pdf/2308.01875.pdf

 

[albie]

I said the infinite size was an extrapolation from measurements, not a measurement itself.

Infinite and flat is our current best fit model, but infinite and negative curvature or finite and closed (but very large) are both possibilities.

You can measure the scale of the fluctuations in the cosmic microwave background. How they look depends slightly on the curvature of spacetime.

>>You can measure the scale of the fluctuations in the cosmic microwave background. How they look depends slightly on the curvature of spacetime.

How much of the universe have we mapped for microwaves? to the limits of the supposed edge or just as far as our instruments allow? Could the universe be flat as far as we have measured and still have an edge?

 

[PeroK]

Could the universe be flat as far as we have measured and still have an edge?

There's no working theory that would allow such a thing. For example, what would happen to a light beam that reaches the edge?

In general, physics isn't about "what could possibly be the case". It's more about proposing a model that satisfies the observed phenomena as simply as possible.

For example, if you were a physics graduate and spent your time trying to find mathematical models where the universe has an edge, you'd be largely wasting your time.

There's also a difference between purposeful speculation (e.g. in proposed theories of quantum gravity) and idle speculation (is there a zoo of pink unicorns at the edge of the universe).

 

[Ibix]

How much of the universe have we mapped for microwaves?

I don't think you understand what the microwave background is.

to the limits of the supposed edge

There is no edge as far as we are aware, so this is meaningless.

just as far as our instruments allow?

The CMB comes from the last scattering surface, which is the time at which the matter in the universe cooled enough to become transparent to electromagnetic radiation. We can't see further than that because light emitted further away was either emitted before the universe became transparent, or too far away to have had time to reach us.

Could the universe be flat as far as we have measured and still have an edge?

Our best answer to this is no. Our models could be wrong, of course, but edges are hugely problematic. Any model that includes an edge would need to answer questions like what having an edge to the universe even means. What happens to things that reach the edge? Why is everything the same everywhere except at the edge? What's beyond the edge? How can we write laws of physics that work in a finite region and forbid anything from existing outside that region? Or are somehow discontinuous at the edge?

It raises way more questions than the zero questions that it answers.

 

[albie]

I don't think you understand what the microwave background is.

There is no edge as far as we are aware, so this is meaningless.

The CMB comes from the last scattering surface, which is the time at which the matter in the universe cooled enough to become transparent to electromagnetic radiation. We can't see further than that because light emitted further away was either emitted before the universe became transparent, or too far away to have had time to reach us.

Our best answer to this is no. Our models could be wrong, of course, but edges are hugely problematic. Any model that includes an edge would need to answer questions like what having an edge to the universe even means. What happens to things that reach the edge? Why is everything the same everywhere except at the edge? What's beyond the edge? How can we write laws of physics that work in a finite region and forbid anything from existing outside that region? Or are somehow discontinuous at the edge?

It raises way more questions than the zero questions that it answers.

>>The CMB comes from the last scattering surface, which is the time at which the matter in the universe cooled enough to become transparent to electromagnetic radiation. We can't see further than that because light emitted further away was either emitted before the universe became transparent, or too far away to have had time to reach us.

So we haven't mapped far enough to determine if there is an edge or curves.

>>

Any model that includes an edge would need to answer questions like what having an edge to the universe even means. What happens to things that reach the edge?

Surely the universe is space time therefore beyond the edge would be literally nothing(no space or time) until matter and energy reach into it and create more space time.
We know this happened with the big bang so why cannot it be a good model?

 

[Ibix]

So we haven't mapped far enough to determine if there is an edge or curves.

If it's actually infinite, you will always be able to make this claim. Adding an edge and all it entails just out of sight isn't wrong per se, but it's unjustified and unfalsifiable, which ought to be a warning.

Surely the universe is space time therefore beyond the edge would be literally nothing(no space or time) until matter and energy reach into it and create more space time.

The point is that "beyond the edge of the universe" isn't necessarily a thing that makes sense, since position and time are concepts provided by spacetime. That's one reason why an edge is difficult to deal with.

We know this happened with the big bang so why cannot it be a good model?

I don't think we do know that. An infinite universe was always infinite in size. And we know that there is a singularity in our naive models that is often described as a "beginning of the universe", but the singularity is actually our models breaking down. So we know that the model is inaccurate, and we can only say that the Big Bang singularity is the beginning of the model and we do not know how the real universe behaved. More sophisticated models avoid the singularity in one way or another, but then there isn't a start to the universe in those models either.

 

[PeroK]

Surely the universe is space time therefore beyond the edge would be literally nothing(no space or time) until matter and energy reach into it and create more space time.
We know this happened with the big bang so why cannot it be a good model?

The Big Bang model is the opposite of what you claim. The model does not involve space expanding into a "void". And, space does not have an edge in the Big Bang model even if the universe is finite.

 

[PeterDonis]

Surely the universe is space time therefore beyond the edge would be literally nothing(no space or time) until matter and energy reach into it and create more space time.

This is nonsense. You can't "create more spacetime". Spacetime already includes time, so it already includes all of the effects that will happen over time.

We know this happened with the big bang

We know no such thing. Where are you getting this nonsense from?

 

[albie]

This is nonsense. You can't "create more spacetime". Spacetime already includes time, so it already includes all of the effects that will happen over time.We know no such thing. Where are you getting this nonsense from?

Why are you being rude to me? By creating more space time I meant advancing the spacetime outwards.

The most common view of the big bang is of a singularity exploding into nothingness.

 

[PeroK]

If there was no void then there must have been an obstacle. By void I mean zero dimensions.

Cosmology has progressed enormously in the past 100 years, from Einstein's General Theory of Relativity to observations of an accelerating expansion, to measuring gravitational waves.

There are unanswered questions, most notably whether there is dark matter influencing galaxy rotation curves and what causes the accelerating expansion.

Modern cosmogy does not rely on meeting with your approval. Cosmologists will continue to research and develop the theories we have. That research is not going to stop and be replaced with "ask albie what he thinks".

 

[vanhees71]

Why are you being rude to me? By creating more space time I meant advancing the spacetime outwards.

The most common view of the big bang is of a singularity exploding into nothingness.

The most common view is that a sufficiently coarse-grained picture of the universe can be described as a Friedeman-Lemaitre-Robertson-Walker spacetime, which has necessarily (Penrose/Hawking) singularities, where the theory breaks down, and nothing can be said about what was really going on close to these singularities.

For the rest of this exact solution of the Einstein field equations, which also implies that the "cosmological substrate" is described as some ideal fluid (which is no surprise given the coarse-graining over large spatial areas we are considering), whose "content" is defined by the corresponding equations of state, there's no "nothingness" into which something explodes. The FLRW solution simply describes the spacetime of the entire universe. The choice of the various components of the matter (+em. radiation) can be inferred from observations. The most important ones are the measurement of the angular distribution of the fluctuations of the temperature of the cosmic microwave background radiation and the distance-redshift relation (Hubble-Lemaitre relation) from measurements on supernovae, which an be taken with some conficence as "standard candles". The best-fit result is then what's described as the "history of our universe":

https://en.wikipedia.org/wiki/Physical_cosmology

 

[Dale]

It's not impossible, no, but it's adding complexity to the model in a way that is deliberately undetectable and with no theoretical justification beyond "it could be true".

It is not impossible, but the resulting models violate Occham’s razor and have less predictive power than current models.

 

[PeterDonis]

Why are you being rude to me?

Pointing out that you are wrong is not rude. It's what you should expect here when you make wrong statements. Particularly when you continue to make wrong statements after it should be abundantly clear to you that your understanding of the subject is flawed and multiple people with much better understanding are trying to help you improve it, but you're not listening to them.

By creating more space time I meant advancing the spacetime outwards.

Which is still wrong.

The most common view of the big bang is of a singularity exploding into nothingness.

Wrong. The Big Bang was not a explosion "into" anything.

 

[PeterDonis]

If there was no void then there must have been an obstacle.

Wrong.

By void I mean zero dimensions.

Wrong.

You need to stop making statements based on your flawed understanding and actually take the time to read, digest, and consider the multiple valid responses you have been given in this thread. Otherwise the discussion will just keep going in circles, which is pointless and a waste of everyone's time.