Shape of Universe: Is Flatness Approved? Causes of Big Crunch

[chill_factor]

Okay thanks.
These triangles you are talking about aren't in one plane, so their angles don't have to add to 180 degrees. It's sad that the shape can't be imagined, as it is beyond our senses' experiences.

Doesn't matter. A plane is just a surface. But a surface is not necessarily a plane. A triangle on a surface that happens to be flat always has angles that add to 180 but a triangle on an arbitrary surface doesn't have to have angles that add to 180.

A surface is just a place where things happen in 2-D. If you were a 2-D creature on a 2-sphere, there is absolutely no way to tell whether you were in a single plane or not; the only way you could tell whether you were on a 2-sphere or on a flat plane is to draw a big triangle and measure the angles.

 

[chasw]

snip... we, as inferior mankind, through our limited perception of the world, marvel at the impossibility of ideas like "entanglement", the "sole beginning bang of infinite time from nothingness" and even the the philosophical contradiction of the theoretical existence of "totally empty space" ...snip.

My thoughts exactly. Ultimately, cosmology is a philosophical problem with inputs from physics. - CW

 

[Chalnoth]

My thoughts exactly. Ultimately, cosmology is a philosophical problem with inputs from physics. - CW

I think you're thinking of cosmogony. Cosmogony is about the origins of the universe. Cosmology is about the evolution of the universe. Cosmology is strongly tied to observation today, and is considered a specific branch of astrophysics. Cosmogony is a bit more speculative, due to a lack of hard data on the subject, and thus most of the arguments regarding it have to be done in the absence of data, which makes the philosophical aspect more important.

That said, all of science uses and relies upon philosophy for its conclusions.

 

[chasw]

Thanks Chalnoth for the correction. I was in fact thinking of the origins of the universe and the highly plausible big bang model. For example, the paradox of rapid inflation, at speeds faster than light, boggles the mind.

It seems humans cannot adequately explain what preceeded creation (nothingness?), what triggered it (prime mover?) and how all that matter and energy unfolded in the first few milliseconds. We are too far removed from these unique events to ever move the dialog beyond speculation.

As far as the rest of the universe's story leading up to the present time and beyond, civilization is dependent on a relatively small number of people who are exploring the universe through ever more sophisticated instruments. Even here, the mysteries never seem to end. - CW

 

[marcus]

I think you're thinking of cosmogony. Cosmogony is about the origins of the universe. Cosmology is about the evolution of the universe. Cosmology is strongly tied to observation today, and is considered a specific branch of astrophysics. Cosmogony is a bit more speculative, due to a lack of hard data on the subject, and thus most of the arguments regarding it have to be done in the absence of data, which makes the philosophical aspect more important.

That said, all of science uses and relies upon philosophy for its conclusions.

I'll chip in here. I completely agree with Chally's statement here. I often disagree with him on lesser details but this sounds exactly right.

I think it's important not to confuse cosmology with cosmogony (investigating the origin of the universe.)

In pop-literature there is a confusion of the start of expansion with the origin of the universe. We do not know that the start of expansion (popularly called "big bang" though not a good description) coincided with the origin of the universe, or with some "creation out of nothing" event.

There are various models that extend time back before start of expansion. There is reason to hope that we may be able to sort this out and select the best fit model.

I would take issue with what you, Charles, say here: I don't think it's such a good idea to equate (as they do in popular media) the conventional big bang model with "origins of the universe". We do not have a scientific basis of evidence for assuming the two are the same. The conventional model simply breaks down as it approaches the start of expansion, so more robust extensions of it are being developed.

... I was in fact thinking of the origins of the universe and the highly plausible big bang model. For example, the paradox of rapid inflation, at speeds faster than light, boggles the mind. ...

Distances increasing at speeds > c are routine. Most of the galaxies one can observe with telescopes are at distances which are increasing faster than the speed of light. Any distance greater than 14 billion lightyears is doing that. The galaxies are not moving thru space to any great extent, the distances between are simply increasing. Geometry is dynamic and we have no right to expect that stationary objects will not gradually become farther apart.
The current rate of expansion is 1/140 of a percent per million years. This does not sound like much but a distance of 14 billion lightyears growing at that rate is growing at the speed of light. A distance twice that would be growing (proportionally) at twice the speed.
This need not boggle anyone's mind (unless there is some wish to be boggled and a desire to tell people about it)---anyway that's my view.
You could watch the balloon model. Shows galaxies staying still (at fixed lat and long.) and getting farther apart faster than the wiggling photons of light (also shown) can travel.

 

[Thomas1989]

As I recall a very recent report, from South Pole Telescope, said that with 95% certainty the curvature was not zero but just a wee bit on the positive side of zero! So that while the U is not infinite (according to them) it is so nearly flat that the hypersphere circumference could be as large as 880 billion lightyears. That is, the 3D analog of a sphere so that if you could stop expansion right now and sail off at light speed in some direction you could travel in a straight line for 880 billion years before you found yourself back home. But it might not be that near flat, or that large--there is a range of uncertainty about the mean curvature.

marcus, I've been reading your posts all day, and while this sort of stuff is certainly far, far beyond my understanding, I'm really enjoying just trying to make sense of them. Mind blowing stuff.

Is there any way you can put this in simpler terms, or any diagrams knocking about that could explain this to a layman like me? Does this report imply the universe is finite and shaped like a sphere and incredibly huge, or am I way off here?

 

[usmhot]

Hi all.
Total noob here, but extremely interested. Only got as far as 2nd year college physics and that was some 20 years ago, so feel free to explain things to me as if I was a 10 year old.

I've been puzzling over recent remarks that the Universe is spatially flat. Earlier in this thread ...

Also keep in mind their is no clear consensus if the universe is open or closed. At this point we can only say that it is flat or extremely close to flat.
As mentioned in a month as Marcus mentioned. We will be getting further data.
The sticky thread on the balloon analogy also has tons of useful links. I highly recommend the ones leading to Ned Wrights tutorials. Particularly his FAQ article. Its one of the better articles for those relatively new to cosmology.
Some things to add on the open closed description. If the universe is closed/finite now then its always finite. Same applies to infinite/open.

... which indicates what seems completely logical to me - that if the Universe is flat then it must be infinite. In fact, if the Universe is topologically open then it must be infinite (right? ... at least according to my understanding of the cosmological principal). So, whether it's flat or saddle shaped (negative curvature?) it must be infinite.

Now, what I don't understand is how a Big Bang Universe can be spatially open and infinite. Surely, a singularity is closed and topologically spherical?

Or, in other words, I can picture where a Big Bang could be on a (hyper)sphere, but not on a saddle or plane.

Can somebody explain this to me (like I'm a 10 year old :) ).

Thanks.

 

[petergreen]

Banana

 

[Chalnoth]

... which indicates what seems completely logical to me - that if the Universe is flat then it must be infinite. In fact, if the Universe is topologically open then it must be infinite (right? ... at least according to my understanding of the cosmological principal).

Not at all. A torus (doughnut shape) is topologically flat, because you can wrap a flat sheet into a torus without tearing or kinking. Visually, living inside a torus-shaped universe would be rather like the classic video game Asteroids.

 

[Naty1]

... which indicates what seems completely logical to me - that if the Universe is flat then it must be infinite.

flat SPACE need not mean flat SPACETIME...

 

[usmhot]

Isn't a torus topologically equivalent to a sphere? Anyway, it's curved and finite and, in particular, unbounded.

Just focused on space though ... flat must be infinite, otherwise it must have boundaries.

So, what shape was the Big Bang? Wasn't it finite and unbounded and closed?

I should have been clearer in my original question ... my problem is with the fact that the Universe must be infinite if it's flat because otherwise it would have to be bounded and would thus break the cosmological principle. So, logically, if it's flat, it must be infinite and it must always be infinite.

 

[Chalnoth]

Isn't a torus topologically equivalent to a sphere?

No. You can't make a sphere out of a flat sheet without tearing. You can make a torus out of a flat sheet without tearing.

So, what shape was the Big Bang? Wasn't it finite and unbounded and closed?

Unknown, and possibly unknowable. We can only see a small slice of the universe, due to the speed of light limitation. We haven't yet definitively detected any overall spatial curvature in our visible patch, but even if we did, that would only tell us about our visible patch. Imagine, for example, that we detect some amount of positive curvature. That could mean our universe is sort of like the surface of a sphere, or it could mean we're living on a sort of hill on a sheet with lots of hills and valleys.

The only way to get at the answer is to learn the correct model for how our universe began, and then get lucky in that model telling us unambiguously what the shape of our universe must be.

I should have been clearer in my original question ... my problem is with the fact that the Universe must be infinite if it's flat because otherwise it would have to be bounded and would thus break the cosmological principle. So, logically, if it's flat, it must be infinite and it must always be infinite.

The cosmological principle is just a simple assumption that is probably wrong when you get to large enough scales.

 

[TrickyDicky]

my problem is with the fact that the Universe must be infinite if it's flat because otherwise it would have to be bounded and would thus break the cosmological principle. So, logically, if it's flat, it must be infinite and it must always be infinite.

As pointed out before the torus is flat, unbounded and finite.
A different thing is that I believe a 3 dim torus can't be embedded in a Lorentzian manifold so I guess that would leave us with the "flat space means infinite space" assumption again.

 

[TrickyDicky]

The cosmological principle is just a simple assumption that is probably wrong when you get to large enough scales.

I don't think it is a simple assumption that is probably wrong, it is "the" assumption that sustains the LCDM model(including the flat space, inflation and dark matter and dark energy assumptions) and GR's FRW metrics.
If it is indeed wrong all that has to be questioned.
are you influenced by the Planck data confirming anomalies at large scales to say that?

 

[Chalnoth]

I don't think it is a simple assumption that is probably wrong, it is "the" assumption that sustains the LCDM model(including the flat space, inflation and dark matter and dark energy assumptions) and GR's FRW metrics.

It only has to hold within the observable universe for this to be the case. Once we start going beyond the observable universe, well, pretty much anything goes. We expect that the cosmological principle must hold significantly beyond the observable universe primarily because if it didn't, we would expect to see some deviation within it as well. But there's no reason to believe it holds out to infinity.

are you influenced by the Planck data confirming anomalies at large scales to say that?

Not at all. I would have told you the exact same thing five years ago.

 

[usmhot]

OK. Had to do some reboning ... in my distant memory I had mixed up the fact that a torus degenerates into a sphere to the belief that it is equivalent, which, of course it isn't.

And, indeed, I understand now how a torus is topologically equivalent to a flat plane, but is a closed unbounded surface. So, a torus (or any topologically equivalent surface?) could be a valid shape for a finite unbounded flat Universe.

I accept that there might be some question of the actual validity of the Cosmological Principle, but, for the sake of discussion, let's assume it holds at the very large scale. Is a torus shaped Universe consistent with the Cosmological Principle? If not, what shape(s) are?

 

[Chalnoth]

OK. Had to do some reboning ... in my distant memory I had mixed up the fact that a torus degenerates into a sphere to the belief that it is equivalent, which, of course it isn't.

And, indeed, I understand now how a torus is topologically equivalent to a flat plane, but is a closed unbounded surface. So, a torus (or any topologically equivalent surface?) could be a valid shape for a finite unbounded flat Universe.

I accept that there might be some question of the actual validity of the Cosmological Principle, but, for the sake of discussion, let's assume it holds at the very large scale. Is a torus shaped Universe consistent with the Cosmological Principle? If not, what shape(s) are?

Almost, but not quite. A torus shape is fundamentally anisotropic, because you can return to your previous position in a rather short distance only if going in very specific directions.

To fully satisfy the cosmological principle, you need a sphere or a plane (or a much more complicated shape for negative curvature).

 

[usmhot]

... A torus shape is fundamentally anisotropic, because you can return to your previous position in a rather short distance only if going in very specific directions.

To fully satisfy the cosmological principle, you need a sphere or a plane (or a much more complicated shape for negative curvature).

Right. That makes sense.

So, if the cosmological principle holds then there are only two possibilites - the Universe is either a sphere or it is infinite?

 

[George Jones]

And, indeed, I understand now how a torus is topologically equivalent to a flat plane

A flat torus is not topologically equivalent to a flat plane. A torus is formed from a plane by topological identifications (i.e. a torus is a quotient space). See my attachment. The identifications change the topology.

A flat plane is simply connected, while a flat plane is not simply connected. Every closed curve in the plane is contractible to a point. A closed curve that loops around the "small" circumference of the torus is not contractible to a point.

A torus is a homogeneous space, but not isotropic. A flat plane is homogeneous and isotropic.

Even though the torus in my attachment looks curved, it is actually flat.

[edit]Chalnoth also posted about this.[/edit]

 

[George Jones]

Almost, but not quite. A torus shape is fundamentally anisotropic, because you can return to your previous position in a rather short distance only if going in very specific directions.

To fully satisfy the cosmological principle, you need a sphere or a plane (or a much more complicated shape for negative curvature).

So, if the cosmological principle holds then there are only two possibilites - the Universe is either a sphere or it is infinite?

Topologically, yes, i.e., the plane and the space of negative curvature used in FLRW universes are topologically equivalent.

 

[usmhot]

Topologically, yes, i.e., the plane and the space of negative curvature used in FLRW universes are topologically equivalent.

OK. Thanks.

So, can a spatially infinite Universe come from a Big Bang?

 

[Chalnoth]

OK. Thanks.

So, can a spatially infinite Universe come from a Big Bang?

Depends upon the model.

But whichever way you slice it, there's still no reason to believe that the cosmological principle holds at scales much larger than our horizon.

 

[usmhot]

Depends upon the model.

But whichever way you slice it, there's still no reason to believe that the cosmological principle holds at scales much larger than our horizon.

Well, there's one reason to suggest it - if doesn't hold everywhere then it's likely not to hold in our observable Universe, right? (As you implied yourself

We expect that the cosmological principle must hold significantly beyond the observable universe primarily because if it didn't, we would expect to see some deviation within it as well.

But, anyway, just exploring some thoughts ... assume the following axioms (as well as the laws and constants as measured in our observable Universe)
1. The Universe started from a Big Bang singularity
2. The Cosmological Principle holds

can you explain how an infinite Universe can exist?

 

[Chalnoth]

Well, there's one reason to suggest it - if doesn't hold everywhere then it's likely not to hold in our observable Universe, right? (As you implied yourself

That's stating it too strongly. Isotropy and homogeneity are likely to hold at scales significantly larger than our horizon, but this doesn't mean that they hold infinitely-far.

I would also like to point out that the cosmological principle is most certainly not a precise description of even our own, visible universe: there are differences in density from place to place.

But, anyway, just exploring some thoughts ... assume the following axioms (as well as the laws and constants as measured in our observable Universe)
1. The Universe started from a Big Bang singularity
2. The Cosmological Principle holds

can you explain how an infinite Universe can exist?

This is insufficient. The problem is that a Big Bang singularity is ill-defined (and also nonsensical).

Also, if you think it's weird that our universe may be infinite in space, bear in mind that it seems to be infinite in time: in the future, our universe is likely to expand forever.

 

[Passionflower]

I don't think it is a simple assumption that is probably wrong, it is "the" assumption that sustains the LCDM model(including the flat space, inflation and dark matter and dark energy assumptions) and GR's FRW metrics.

Until I see some mathematical proof that a mostly void space with extremely small but massive lumps of matter can accurately be modeled by an FRW metric I would stay skeptical.

 

[usmhot]

That's stating it too strongly. Isotropy and homogeneity are likely to hold at scales significantly larger than our horizon, but this doesn't mean that they hold infinitely-far.

[snip]

Also, if you think it's weird that our universe may be infinite in space, bear in mind that it seems to be infinite in time: in the future, our universe is likely to expand forever.

I don't think it's weird that it may be infinite in space ... I think it's impossible. Not least because infinity is not a number.

And, in fact, it's good that you said that it seems to be infinite in time, because that exemplifies the problem with using 'infinity'. The Universe will not exist for an infinite time. At any point at which one would care to measure it no matter how far in the future the measurement will be a finite number. There is an important difference between existing for an indefinite amount of time and existing for an infinite amount of time.

So, to get back to its size - if it was ever finite in size then it will always be finite in size, as, no matter how much space you add to it, as long as you add a finite amount it will still be finite. Which implies that if it is infinite in size now then when it came into existence it must have been infinite in size. Which, I believe, is technically absurd. (Which is why I’d appreciate an explanation of a model in which an ‘infinite’ Universe can come from a Big Bang.)

So, the Cosmological Principle is extremely important. If it holds, then, as far as I can see, the only possible topology is a sphere (albeit with such a large radius that it is close to flat on the scale that we can measure). But, if the Cosmological Principle does not hold then either the Universe is a torus or it has boundaries.

 

[Chalnoth]

I don't think it's weird that it may be infinite in space ... I think it's impossible. Not least because infinity is not a number.

Infinity is a number on the extended number line. It's a slightly weird number, but it is a number.

And, in fact, it's good that you said that it seems to be infinite in time, because that exemplifies the problem with using 'infinity'. The Universe will not exist for an infinite time. At any point at which one would care to measure it no matter how far in the future the measurement will be a finite number. There is an important difference between existing for an indefinite amount of time and existing for an infinite amount of time.

This is playing word games. The dimension of time for our universe is likely to be infinite in extent.

So, to get back to its size - if it was ever finite in size

Why do you think it was ever finite in size?

So, the Cosmological Principle is extremely important. If it holds, then, as far as I can see, the only possible topology is a sphere (albeit with such a large radius that it is close to flat on the scale that we can measure). But, if the Cosmological Principle does not hold then either the Universe is a torus or it has boundaries.

Why not some sort of irregular blobby shape?

 

[Chronos]

I think the answer to the flatness question is undefinable. We know it is practically zero, but, will never never know if it is perfectly flat. I prefer to think it fluctuates around zero, but, is never exactly zero due to quantum uncertainty.

 

[Chalnoth]

I think the answer to the flatness question is undefinable. We know it is practically zero, but, will never never know if it is perfectly flat. I prefer to think it fluctuates around zero, but, is never exactly zero due to quantum uncertainty.

Or put another way, it can only be measured if it is significantly non-zero.

 

[TrickyDicky]

Infinity is a number on the extended number line. It's a slightly weird number, but it is a number.

Infinity is not considered a number(weird or not) in mathematics. It's more like a concept

Why do you think it was ever finite in size?

Well, usmhot has a point there, if it was infinite from the first instant after t=0, it doesn't make much sense to talk about changes of spatial size, or inflationary epochs, IOW how is something infinite comparable in size at different times, it seems logical that it would be equally infinite everytime.

 

[Chalnoth]

Well, usmhot has a point there, if it was infinite from the first instant after t=0, it doesn't make much sense to talk about changes of spatial size, or inflationary epochs, IOW how is something infinite comparable in size at different times, it seems logical that it would be equally infinite everytime.

It isn't clear at all that there was an absolute beginning, before which there was nothing. And certainly there was no singularity.

Furthermore, changes in size are not done with regard to the whole, but with regard to changes of distance within the universe. There is no problem whatsoever for an infinite universe to expand: it means that average distances between things in the universe are getting larger.

I would like to point out that the flat FRW metric that is generally used to examine these things is infinite in extent.

 

[George Jones]

Well, usmhot has a point there, if it was infinite from the first instant after t=0, it doesn't make much sense to talk about changes of spatial size

Spatial scale is not determined by topology (i.e., whether space is compact or non-compact), it is determined by an additional structure, the metric (as noted by Chalnoth). FLRW universes have time-dependent metrics (no timelike Killing vectors).

 

[chill_factor]

Infinity is not considered a number(weird or not) in mathematics. It's more like a concept

Sure it is. Make a graph of numbers where the X axis is defined by 1/R.

(0,0) is then infinity defined to be a single point on this graph.

 

[WannabeNewton]

Sure it is. Make a graph of numbers where the X axis is defined by 1/R.

(0,0) is then infinity defined to be a single point on this graph.

What?

I repeat...what?? I don't even...O.O

 

[Chalnoth]

Sure it is. Make a graph of numbers where the X axis is defined by 1/R.

(0,0) is then infinity defined to be a single point on this graph.

This isn't true. The value at 0 in such a graph is undefined. This can be understood as due to the fact that if you take the limit as x approaches 0 for 1/x, you get different answers if you approach zero from the positive direction vs. the negative direction ($+\infty$ and $-\infty$, respectively).

Anyway, what you call infinity is somewhat irrelevant. It does behave differently from other numbers in a few fundamental ways (that is, it behaves differently under various operations than other numbers). But the fact of the matter is, none of this has any bearing on whether or not the concept of infinity can be applied to reality. Even if we claim that space-time is fully-described by the real numbers and not the extended reals, and if our space-time maps onto all of the reals, then it is infinite in extent (in both time and space). There is nothing nonsensical about this statement.