More on Infinity: Why Does Balloon Analogy Not Describe Universe?

[Pjpic]

Why does the balloon analogy not accurately describe the universe?

Common sense would say that an expanding isomorphic homogenous space would have to be curved in a higher, infinite, spatial dimension.

Astronomers seem unsure if the universe is finite or infinite.
Physicists seem to, according to wiki, wish to avoid the concept of infinity.
Mathematicians are a little opaque, for me, on most issues; so, I don't know if they are saying infinity can expand.


Can anyone tell me what the "official" teaching on the balloon analogy is?

 

[epkid08]

Why does the balloon analogy not accurately describe the universe?

Common sense would say that an expanding isomorphic homogenous space would have to be curved in a higher, infinite, spatial dimension.

Astronomers seem unsure if the universe is finite or infinite.
Physicists seem to, according to wiki, wish to avoid the concept of infinity.
Mathematicians are a little opaque, for me, on most issues; so, I don't know if they are saying infinity can expand.


Can anyone tell me what the "official" teaching on the balloon analogy is?

I would say that the notion of infinity expanding goes against the definition of infinity. I think it's quite obvious that the three spatial dimensions we are bound to are finite. Although, higher dimensions, like the one the balloon model suggests, may or may not be finite; who knows, maybe they even have the predisposition of being infinite.

 

[Pjpic]

I would say that the notion of infinity expanding goes against the definition of infinity. I think it's quite obvious that the three spatial dimensions we are bound to are finite. Although, higher dimensions, like the one the balloon model suggests, may or may not be finite; who knows, maybe they even have the predisposition of being infinite.

I suppose something would either be infinite or have a boundary beyond which there is nothing.

 

[Hurkyl]

Common sense would say that an expanding isomorphic homogenous space would have to be curved in a higher, infinite, spatial dimension.

How could it, or anything else on this topic, possibly be common sense?

 

[Chronos]

The universe is not constrained by the limits of human comprehension.

 

[HallsofIvy]

I suppose something would either be infinite or have a boundary beyond which there is nothing.

No, that's not true. To use, again, the "balloon" analogy, the surface area of sphere is finite but has no boundary.

If the universe has positive overall curvature, then it could be finite without having a boundary.

 

[marcus]

Why does the balloon analogy not accurately describe the universe?

Common sense would say that an expanding isomorphic homogenous space would have to be curved in a higher, infinite, spatial dimension.

Astronomers seem unsure if the universe is finite or infinite.
Physicists seem to, according to wiki, wish to avoid the concept of infinity.
Mathematicians are a little opaque, for me, on most issues; so, I don't know if they are saying infinity can expand.
...

You sound as if you've already made a pretty good start towards understanding this business.

You probably understand the point that HallsofIvy just made----a balloon surface is finite area.

You may be going to get tripped up by the idea of intrinsic curvature. Ever since around 1850 it has been possible to understand how a surface can be curved without living in higher dimensional surroundings. You can test curvature by measuring the angles of a triangle, or testing the pythagoras right triangle rule, or various other ways.

We could be living in the 3D analog of a balloon surface and measure the curvature and yet there might not be a surrounding space. Our 3D space could be the only space there is and yet it could be curved. This (in my experience at PF cosmology forum) is probably what newcomers find hardest to grasp.

The thing to remember is that we have no right to expect that the angles of a triangle add up to 180 degrees. Maybe with very large triangles they add up to more! It is the case with Euclidian geometry that they add up to 180 degrees. But we have no guarantee that the geometry of our world is Euclidian.
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About what you said about physicists not liking infinity----you have to be careful what you mean here. There are two ideas of infinity: infinite spatial extent and infinite volume is a harmless commonplace type of infinity. It is one possibility in cosmology and in my experience physicists and everybody else are comfortable with it.
The other kinds of infinities are where a mathematical model breaks down or blows up and stops computing reasonable numbers. then it is just broken and it is time to fix it. Say some model starts giving answers like in some bounded region there is infinite energy or infinite density or infinite radiant power or electric charge. That would be a singularity (a mathematical breakdown of the theory).

If you are still bothered by the idea that physicists don't like infinities, that you found in Wikipedia, you could bring a link to the Wikipedia article and we could look at it. They probably are not talking about infinities like infinite spatial extent or volume---it's probably the other kind.
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The most recent WMAP report (the 5 year CMB data, implications for cosmology) gave a lowerbound estimate for the radius of curvature of space.

They said it was at least 100 billion lightyears, if I remember correctly---some estimate of that magnitude. At least means it could be infinite too, which would be the flat Euclidian case. Anyway a finite but very large RoC fits in with the balloon picture you mentioned. If you want a link to that WMAP report let us know. I think it's neat that they gave an estimate.

 

[Pjpic]

No, that's not true. To use, again, the "balloon" analogy, the surface area of sphere is finite but has no boundary.

If the universe has positive overall curvature, then it could be finite without having a boundary.

Wouldn't there be a boundary between the surface and what is "above" the surface?

 

[Pjpic]

The universe is not constrained by the limits of human comprehension.

And the limit of human comprehension is what?

 

[Pjpic]

How could it, or anything else on this topic, possibly be common sense?

Because the balloon analogy is common sensical.

 

[Pjpic]

We could be living in the 3D analog of a balloon surface and measure the curvature and yet there might not be a surrounding space. Our 3D space could be the only space there is and yet it could be curved. This (in my experience at PF cosmology forum) is probably what newcomers find hardest to grasp.

This is what I'm not understanding. I can see how there may be nothing beyond our space but wouldn't that mean there is a boundary between our space and nothing (or what might not be nothing)?

 

[marcus]

Indeed there COULD be some surrounding higher dim'l space---though we have no evidence of it so far.

And there also might not be. To repeat: Our 3D space could be the only space there is.
In that case there is obviously no boundary.
I can't say it any clearer Pj, no boundary because no other space---you either get it or you don't.

 

[Pjpic]

Indeed there COULD be some surrounding higher dim'l space---though we have no evidence of it so far.

And there also might not be. To repeat: Our 3D space could be the only space there is.
In that case there is obviously no boundary.
I can't say it any clearer Pj, no boundary because no other space---you either get it or you don't.

I wonder what book or article would explain this. It sounds like a contradiction in terms if the "offical" teaching is that something is finite but has no end point (infinite).

 

[MeJennifer]

In general relativity the question is a simple one, spacetime is either open or closed and that implies that the universe is finite if spacetime is closed and infinite if spacetime is open.

 

[Pjpic]

In general relativity the question is a simple one, spacetime is either open or closed and that implies that the universe is finite if spacetime is closed and infinite if spacetime is open.

Is the spacetime concept of a closed universe mean that there is a boundary in time (the big crunch)? But I'm still confused as to where the boundary in space would be.

 

[robertm]

Is the spacetime concept of a closed universe mean that there is a boundary in time (the big crunch)? But I'm still confused as to where the boundary in space would be.

Boundaries have no meaning in a boundless system. Weather infinite or not, curved, closed, or flat, the picture seems to be one of no "boundaries in space".

"The analogue of the two-dimensional sphere is called a three-sphere. In the rather unlikely event of four-dimensional creatures existing, they would be able to visualize the curvature of three-dimensional space in just the same way we can visualize that of two-dimensional space. However, like the two-dimensional sphere, the possible curvature of three-space is an intrinsic property and there is no actual need of a higher-dimensional space for it to live in. Obtaining a correct mental picture of this is one of the big challenges in understanding our Universe!
A Universe with a spherical geometry, like the surface of the Earth, has a finite size but no boundary. All points are equivalent. If we live in a spherical geometry, and travel in a straight line, we would not go on for ever and ever. Rather, eventually we would come back to where we had started from, from the opposite direction, exactly in the manner that someone traveling outward from the North Pole on the Earth eventually returns there from the opposite direction." An introduction to Modern Cosmology 2nd ed, Liddle

So even if the geometry of the universe turned out to be one of finite scope (the 'closed' positive curvature case), the concept of boundaries, on a cosmologic scale, seems to be omitted from reality.

 

[Pjpic]

the possible curvature of three-space is an intrinsic property and there is no actual need of a higher-dimensional space for it to live in


This is the part I don't understand. Is there an accessible, to a lay person, literature on the subject?

I can see how space could be curved like the iso-bars on a weather map without recourse to a fourth dimension. But I don't see how a finte object can be without boundary without recourse to a higher dimesion.

 

[Chronos]

Unbounded, but, finite - in the words of Stephen Hawking. It is well known the universe is observationally finite [surface of last scattering]. No experiment capable of discerning whether it is 'spatially' finite has been proposed and executed.

 

[robertm]

This is the part I don't understand. Is there an accessible, to a lay person, literature on the subject?

It is an intrinsic mathematical consequence, that being the maths of the theory of General Relativity. Not too much tough stuff, but does rely on differential calculus.

I suppose this would be your best bet for a good cursory introduction. He also gives some links to other materials for further study:

http://www.astro.ucla.edu/~wright/relatvty.htm"

 

[Pjpic]

However, like the two-dimensional sphere, the possible curvature of three-space is an intrinsic property and there is no actual need of a higher-dimensional space for it to live in. Obtaining a correct mental picture of this is one of the big challenges in understanding our Universe!

This is the understanding I'm trying to grasp. I understand 'intrinsic' to mean that the (for example) sadle shape of universe can be discovered by people within the universe without referencing a higher dimension (because, for example, angles in a triangle don't add up to 180).

What I don't understand is how there can be a curved space without being embedding in a higher dimension (like the balloon analogy).

 

[PhanthomJay]

What I don't understand is how there can be a curved space without being embedding in a higher dimension (like the balloon analogy).

I am in full agreement with you. The 2D curved surface of a balloon, although finite and without edge, is bounded , both within and without, by the third (and higher order)dimension. That is not common sense; it is fact. It's curvature could not exist without a third dimension; it's curvature is into the third dimension. Now whether that analogy holds for our multidimensioned universe, is I guess open to question. However, the curvature of light in 3-space, in the presence of mass, has been measured at least once during the famous experiment carried out during a solar eclipse, where such curvature from light of a star was observed and measured. My question is: was that curvature into the 4th spatial dimension? Or just an ordinary ho-hum curve in the boring 3rd dimension? I think not.

 

[yuiop]

What I don't understand is how there can be a curved space without being embedding in a higher dimension (like the balloon analogy).

I am in full agreement with you. The 2D curved surface of a balloon, although finite and without edge, is bounded , both within and without, by the third (and higher order)dimension. That is not common sense; it is fact. It's curvature could not exist without a third dimension; it's curvature is into the third dimension. Now whether that analogy holds for our multidimensioned universe, is I guess open to question. However, the curvature of light in 3-space, in the presence of mass, has been measured at least once during the famous experiment carried out during a solar eclipse, where such curvature from light of a star was observed and measured. My question is: was that curvature into the 4th spatial dimension? Or just an ordinary ho-hum curve in the boring 3rd dimension? I think not.

There is a way that the time dimension acts as the "4th spatial dimension". Differences in the way time advances in different regions of space brings about a curvature that appears like an extra spatial dimension. The role of time as a 4th spatial dimension can be seen in the mathematical equations of relativity. It is easier to visualise this in the analogy of 2D creatures living on a 2D surface. With a bit of imagination, you could place the 2D creatures on a circular piece of paper that is perfectly flat (in 3D), and by placing suitable time dilation on the concentric circles you could convince the 2D creatures by every measurement that they can make in 2 dimensions, that they are living on a 3D sphere. In the same way, it is a matter of personal taste whether you choose to see the universe as 3 spatial dimensions + 1 time dimension or as 3 visible spatial dimensions + 1 invisible spatial dimension. Mathematically they are the same thing. Everything that can be explained by an additional invisible spatial dimension can be explained by time dilation and length contraction.

 

[Chronos]

A concise explanation of unbounded surfaces is not trivial. It requires very complex geometrical concepts to convey with mathematical precision. The common sense, logical explanation you are seeking, Pjpic, does not exist. Analogies are good, but, logically imprecise - as you have deduced. That defect does not exist in the more rigorous mathematics behind them.

 

[PhanthomJay]

In the same way, it is a matter of personal taste whether you choose to see the universe as 3 spatial dimensions + 1 time dimension or as 3 visible spatial dimensions + 1 invisible spatial dimension. Mathematically they are the same thing. Everything that can be explained by an additional invisible spatial dimension can be explained by time dilation and length contraction.

Kev, that is a very helpful explanation, thanks. But, what about M-Theories 5th and 6th etc. extra dimensions? Are they postulated to be time-like or space-like? Or both? Could not our 4D spacetime universe be curved into these dimensions?

 

[ThomasT]

Because the balloon analogy is common sensical.

I don't see how the balloon analogy is common sensical. If one assumes some form of a Big Bang conjecture wrt the origin of our universe, then common sense would seem to dictate that our universe is some sort of expanding wave structure, and that the contents (the flotsam and jetsam created in the wake of the expanding universal wave front) of that expanding wave structure is bounded by an expanding universal wave front.

We, and all other material phenomena that we collectively refer to as our universe, are the inside of the isotropically expanding universal wave structure. The surface (and beyond) of this universal wave structure would seem to be necessarily off limits to us or anything else that originated inside it because the rate of expansion would circumscribe an absolute limit wrt the rate of propagation of any disturbance. This is my grossly oversimplified, common sensically speculative view of what our universe is (it's not meant as an analogy).

I realize that the balloon analogy is offered as one way to reify (or at least visualize) the idea of curved space or spacetime which arises via a geometric interpretation of general relativity.

But one doesn't have to take this interpretation literally. More likely, in my common sensical view, GM is itself a gross simplification of the deep reality of gravitational behavior -- which might eventually be modeled in terms of complex wave interactions.

 

[marcus]

I don't see how the balloon analogy is common sensical. .

that could be because you haven't understood the balloon analogy. In the analogy there is no existence where air is----either inside or outside surrounding the balloon.
the only thing that exists is the balloon. That is why it is an analog for the universe.

All that exists is the balloon surface---an idealized 2D world----and the objects and inhabitants are 2D beings in this 2D surface space. That is the analogy.

In your discussion you seem to be imagining a 3D world with an expanding spherical wavefront, and stuff inside of that----a solid ball. That is not the balloon analogy. As far as I can see what you picture does indeed makes no sense, common or otherwise, as an analogy for the universe. Take another try at understanding the balloon analogy. It is actually a pretty good one, and a lot of people find it understandable on a commonsense basis.

the real universe is the 3D analog of the 2D surface of the balloon---with a little work almost anyone can get it, Thomas.

 

[robertm]

with a little work almost anyone can get it

And if you are searching for more than what has already been said, that work will by necessity involve the mathematics of the general theory of relativity.

 

[ThomasT]

that could be because you haven't understood the balloon analogy. In the analogy there is no existence where air is----either inside or outside surrounding the balloon.
the only thing that exists is the balloon. That is why it is an analog for the universe.

I think I understand the balloon analogy. I don't think it's a common sense (or a very good) analogy for our universe. I prefer the raisin bread analogy.

All that exists is the balloon surface---an idealized 2D world----and the objects and inhabitants are 2D beings in this 2D surface space. That is the analogy.

Yes, I understand. But if space isn't actually curved (a la a particular geometric understanding of the deep nature of gravitational behavior), then the balloon analogy is obviated, isn't it?

The reason that cosmologists speak of curved space at all comes from GR, doesn't it?
Can't GR be used without reifying its curved space geometric representation? That is, current (working) gravitational theories don't really say anything about what the fabric of space is -- just that there is one, and, vis GR, at large cosmological scales it's curved. Is this correct?

I don't think that GR is a realistic theory. It's geometry is a (necessary for calculation?) simplification of the deep nature of gravitational behavior, which, my common sense tells me, isn't curved space but complex wave interactions.

In your discussion you seem to be imagining a 3D world with an expanding spherical wavefront, and stuff inside of that----a solid ball. That is not the balloon analogy.

Right, it's the raisin bread analogy -- which I think is much more common sensical, based on what I know of astronomical observations, than the balloon analogy.

As far as I can see what you picture does indeed makes no sense, common or otherwise, as an analogy for the universe.

What I visualize, as I mentioned, was not meant as an analogy per se (at least not in the way that the expanding 2D surface of a balloon is an analogy for an expanding 3D, curved space volume).

Based on observations, some time in the very distant past there was a Big Bang, an incomprehensibly huge disturbance of some sort, a humongous explosion ... right? This was the beginning of our universe.

While this explosion, and resulting disturbance(s), happened (and continue to evolve) in a homogenous, fundamental medium of unknown and unknowable structure, there's no reason to believe that the wave physics that might be used to represent behavior on cosmological (and submicroscopic, for that matter) scales is essentially different than the wave mechanics that is useful on the macroscopic scale that circumscribes our sensory experience.

One can (ideally) picture our universe as an isotropically expanding sphere whose leading edge (boundary) is the shock wave front created during the Big Bang.

We, and the rest of the accessible universe, are inside the boundary of the expanding sphere. We can never breach this boundary -- not because space is curved, but because the universe is expanding. The rate of expansion might vary, but whatever it is during any particular epoch sets the speed limit for the propagation of disturbances within our universe.

Everything is wave behavior. Particles arise from wave interactions. Produce enough particles and you have a particulate medium. The physical universe, the universe that is amenable to our sensory apprehension, including gravitational behavior, is the interfacing and interaction of waves of the various particulate media. Of course, the deep nature of the physical universe isn't amenable to our sensory apprehension, so we have notions like instantaneous-action-at-a-distance and curved space to contend with.

Detailed wave models of reality aren't possible yet, but it is possible to recognize that the idea of a curved space and the use of a balloon analogy to help visualize it are only necessary if one takes the current models of nature (eg., GR) as literal descriptions of it. They aren't.

Take another try at understanding the balloon analogy. It is actually a pretty good one, and a lot of people find it understandable on a commonsense basis.

I do understand it. I just don't think it's a good one -- whether we want to call it common sensical or not.

the real universe is the 3D analog of the 2D surface of the balloon ...

I don't think so. What's being attributed to curved space now will eventually be explained in terms of wave interactions.

Thanks for your response -- I do read most all of the stuff you post. On this thing, I must stubbornly refuse to accept curved space as real until a more compelling argument than the geometry of GR is put forth.

 

[MeJennifer]

Hi Thomas you say repeatedly you understand but reading your posting I believe you do not understand some things.

Can't GR be used without reifying its curved space geometric representation?

It can, but you still have to deal with things like closed spacetimes.

Based on observations, some time in the very distant past there was a Big Bang, an incomprehensibly huge disturbance of some sort, a humongous explosion ... right? This was the beginning of our universe.

I think you would do good in checking some literature as to what the big bang actually is, you are clearly misinformed.

One can (ideally) picture our universe as an isotropically expanding sphere whose leading edge (boundary) is the shock wave front created during the Big Bang.

There is no such thing as a shock wave front, you are misinformed.

We, and the rest of the accessible universe, are inside the boundary of the expanding sphere. We can never breach this boundary -- not because space is curved, but because the universe is expanding.

Even if there is no expansion and spacetime is not closed this is true. Since each observer, that has mass, is within the confines of the expanding future light cone.

 

[ThomasT]

It can, but you still have to deal with things like closed spacetimes.

I choose not to deal with things like that, but if it works for you ...

I think you would do good in checking some literature as to what the big bang actually is, you are clearly misinformed.

The Big Bang was an explosion. You can say that a singularity or some finite volume of space spontaneously started expanding during some time in the distant past, but that's just another way of saying that our universe began with an explosion.

And, explosions produce shock waves.

There is no such thing as a shock wave front, you are misinformed.

And, of course there are such things as shock wave fronts.