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Effort to get us all on the same page (balloon analogy)

by marcus
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Larkus
#451
Dec7-12, 07:03 PM
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Quote Quote by marcus View Post
I would say try to think of the EXPERIENCE of being 2D and living in a 2D sphere. Don't picture the sphere as if you are a God, outside and looking from outside at the sphere. Using some new type of lightrays that travel in 3D rather than 2D. Think of a sphere as the experience of living in it. And also think of a hypersphere that way.

Let's say that you and your brothers discover a remarkable fact about the world namely that there is a special area K which you have determined experimentally which allows you to reliably predict the area of any triangle!
You just have to sum the angles, subtract π, and multiply by that area K!
this always turns out to give the area (if you take the trouble to measure the area carefully.

The rule used by Euclid, namely 1/2 the base times the height does not work for you, it is only approximately right for small area triangles and gets progressively wronger for larger ones.

That's part of what I mean by the experience. It would apply also to living in a hypersphere. It does not involve postulating an extra dimension which we don't experience and cannot access. It just involves experimenting with triangles and determining the value of the area K.

Circumnavigating is another aspect of the experience which you (as creature living in sphere or hypersphere) might have. You can think of various others.

This comic does a good job visualising it in an entertaining way:

'The Adventures of Archibald Higgins: Here's Looking at Euclid'
marcus
#452
Dec18-12, 11:45 AM
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Quote Quote by Larkus View Post
This comic does a good job visualising it in an entertaining way:

'The Adventures of Archibald Higgins: Here's Looking at Euclid'
That's a really nice piece of work! Thanks for the link, Larkus.

I hope you keep us informed about more clever cosmology stuff from Petit and his "learning without boundaries" project.

Today in the "How to prove the stretching of space" thread, I noticed a neat explanation by Brian Powell of how the wavelengths of light get stretched out as distances expand.
Quote Quote by bapowell View Post
From general relativity (specifically, the geodesic equation), it is seen that the momentum of a particle is inversely proportional to the expansion (the scale factor, a(t)). From de Broglie, this becomes a statement about the wavelength of photons -- as space expands, the wavelength of light must increase.
Timmdeeg's reaction says it:
"...your explanation why λ goes with a(t) is very convincing and new to me, thanks."
I think this is an especially nice way to look at it, which doesn't exclude others as well.
marcus
#453
Dec20-12, 04:14 PM
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The South Pole Telescope (SPT) has given us new narrowed-down ranges for the cosmological parameters.

At the highest confidence level these correspond to a cosmos which is NOT "Euclidean flat" and NOT spatially infinite but is the 3D hypersphere analog of the 2D spherical balloon surface model.

The SPT curvature estimates translate into an estimated range of the "radius of curvature" namely from 140 to 320 billion light years.

This may not be right, the U may not be spatially finite, or it might be finite and these numbers might subsequently be revised. But let's take them at face value and see. After all it is a fine instrument, a respected team, and these are the most recent published estimates. Here's what I posted earlier about it:
==quote post #448==
http://arxiv.org/pdf/1210.7231v1.pdf
Scroll to Table 3 on page 12 and look at the rightmost column which combines the most data:
ΩΛ     0.7152  0.0098
H0     69.62  0.79
σ8     0.823  0.015
zEQ    3301  47
Perhaps the most remarkable thing is the tilt towards positive overall curvature, corresponding to a negative value of Ωk

For that, see equation (21) on page 14
Ωk =−0.00590.0040.
Basically they are saying that with high probability you are looking at a spatial finite slight positive curvature. The flattest it could be IOW is 0.0019, with
Ωtotal = 1.0019
And a radius of curvature 14/sqrt(.0019) ≈ 320 billion LY.
Plus they are saying Omega total COULD be as high as 1.0099 which would mean
radius of curvature 14/sqrt(.0099) ≈ 140 billion LY.

For Jorrie's A27 calculator the important parameters as estimated by the SPT report are
current Hubble time = 14.0 billion years
future Hubble time = 16.6 billion years
matter radiation balance Seq = 3300
==endquote==

Since I posted that, Jorrie upgraded calculator from A25 to A27, so I made that change in the quote.

2 pi ≈ 6
so you can, if you wish, estimate the CIRCUMFERENCE of the universe simply by multiplying the "radius of curvature" figures by 6. The smallest it could be is 140 x 6 billion lightyears
and the largest it could be is 320 x 6 billion lightyears.
So if you could stop the expansion process, to make circumnavigation possible, you would have to travel in a straight line for six times 140-320 Gly before you'd be back at starting point.
If you sent a laser flash off in some direction it would be six times 140-320 billion years before it came back at you from the opposite direction.

This is just a way of understanding equation (21) on page 14 of the SPT report.

Ωk =−0.00590.0040.

It's a way to get an intuitive feel in your imagination for what it means.
Here, again, is the link to the technical paper itself:
http://arxiv.org/abs/1210.7231
RayYates
#454
Dec20-12, 05:48 PM
P: 47
In non technical language.. OMG the scale is mind-blowing!
TalonD
#455
Jan11-13, 09:36 PM
P: 177
That's a very disappointing development. Infinite would have been much more aesthetically pleasing to me. Oh well.
marcus
#456
Jan11-13, 10:07 PM
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Quote Quote by TalonD View Post
That's a very disappointing development. Infinite would have been much more aesthetically pleasing to me. Oh well.
Sorry about your sense of disappointment, but hey! it's not settled yet! Uncertainty about that could last a decade, or a generation.

I love the hypersphere S3 the threedimensional analog of the surface of a balloon, so I'm certainly pleased by the South Pole Telescope report, but I have no sense that the thing is finally decided.

But for the sake of an example, if we take the SPT findings at face value then (with 95% confidence) the SMALLEST the circumference could be is 6 times 140 billion LY.
In other words 840 billion light years. quite a big balloon, so to speak. Would take an awfully long time to circumnavigate, if you could stop it from expanding so that circumnavigation would be possible.
TalonD
#457
Jan13-13, 02:18 PM
P: 177
So it's not carved in stone yet? There is still hope for infinity? :D YAY
Mordred
#458
Jan14-13, 12:34 AM
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I have a lot of catching up to do on this thread. Thus far the info contained in it has been insightful. Creedos to Marcus on it. I look forward to the finalized draft.

That being said I found the suggestion of thinking that inside the balloon being the past and outside the future useful. The one concern I have with it is in the case of Black holes. The analogy may lead to misconception that due to its infinite density the singularity may reside in the past at the big bang. I know thats not likely lol but its often the way laymen like myself tend to misconstrue analogies.
Jorrie
#459
Jan14-13, 01:04 AM
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Quote Quote by Mordred View Post
That being said I found the suggestion of thinking that inside the balloon being the past and outside the future useful. The one concern I have with it is in the case of Black holes. The analogy may lead to misconception that due to its infinite density the singularity may reside in the past at the big bang.
Yes, it is a rather troublesome way of viewing the analogy: there could have been a contracting phase in the distant past, followed by a 'bounce'. During such a phase, the past would have been 'outside' and the future 'inside' the balloon.

In any case, if the cosmos happens to be spatially flat or slightly hyperbolic, there can't be a notion of 'inside' or 'outside''. However, the balloon analogy would still yield all the correct answers by just considering the observable universe as the surface patch 'visible' to us.

The motto seems to be: use the analogy to get our brains around the expansion/contraction issue of the surface; then ignore it and rather use the simple mathematics of the LCDM cosmic model (or use one of the many available calculators to play around).
--
Regards
Jorrie
GeorgeDishman
#460
Jan17-13, 06:15 PM
P: 255
Quote Quote by marcus View Post
Sorry about your sense of disappointment, but hey! it's not settled yet! Uncertainty about that could last a decade, or a generation.
I think it is significant that all the results in Fig. 7 are consistent zero curvature other than those that include the BAO result, and BAO is significant in Figure 8 as well. Given the poor agreement between H0 and BAO, I would prefer to resolve the discrepancy before we claim we have strong evidence either way.
marcus
#461
Feb20-13, 10:40 PM
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I don't think I posted anything about the WMAP9 report (Hinshaw et al.) what it said about Ωk . I'll get a link.
http://arxiv.org/abs/1212.5226
Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Parameter Results
G. Hinshaw, D. Larson, E. Komatsu, D. N. Spergel, C. L. Bennett, J. Dunkley, M. R. Nolta, M. Halpern, R. S. Hill, N. Odegard, L. Page, K. M. Smith, J. L. Weiland, B. Gold, N. Jarosik, A. Kogut, M. Limon, S. S. Meyer, G. S. Tucker, E. Wollack, E. L. Wright
(Submitted on 20 Dec 2012 (v1), last revised 30 Jan 2013 (this version, v2))

On page 19 you see:
Ωk= −0.0027+0.0039-0.0038 this is using pretty much all the major data sets: WMAP +eCMB+BAO+H
eCMB or "externalCMB" includes SPT but not the LATEST SPT. But that is details.
When you translate their plus/minus stuff it leads to a confidence interval of
-0.0065 < Ωk < 0.0012

So that is lopsided on the negative Ωk side, which means FINITE but it also has some zero and positive territory which means spatial INFINITE. Thats how several recent major reports have been going. You can't exclude spatial infinite, at this point.

On page 20 they also have a figure −0.0065 0.0040 which I don't take as seriously but which ostensibly is based on even more data namely WMAP +eCMB+BAO+H + SNe.
That would correspond to a purely negative interval:
-0.0105 < Ωk < -0.0025
That would exclude the spatial infinite case, at whatever the confidence level is. But this ball is still up in the air.

Hinshaw et al also had other interesting stuff about other issues, like the number of neutrino species and what Dark Matter clouds might possibly consist of. That tended to get people's attention so what they had to say about curvature was less noticed.
RayYates
#462
Feb21-13, 10:33 AM
P: 47
Quote Quote by Jorrie View Post
Yes, it is a rather troublesome way of viewing the analogy: there could have been a contracting phase in the distant past, followed by a 'bounce'. During such a phase, the past would have been 'outside' and the future 'inside' the balloon.

In any case, if the cosmos happens to be spatially flat or slightly hyperbolic, there can't be a notion of 'inside' or 'outside''. However, the balloon analogy would still yield all the correct answers by just considering the observable universe as the surface patch 'visible' to us.

The motto seems to be: use the analogy to get our brains around the expansion/contraction issue of the surface; then ignore it and rather use the simple mathematics of the LCDM cosmic model (or use one of the many available calculators to play around).
--
Regards
Jorrie
I may have been one of those that suggested looking at the inside as the past and the outside as the future. This was in response to someone trying to calculate the center of the balloon (aka the universe). It was a way of getting him to visualize that everything in the universe is on the surface, and there is no center on the surface.

The balloon analogy also assumes the balloon is always expanding, which as you point out may not be the case.

As for Black Holes, since time and space are the same fabric and space seems to have been compressed out of existence, time must also have stopped, (or nearly so). In my mind I see this as a point/line extending out away from the balloon into infinity (the future). So in a way, its not part of the balloon but still attached at a single point.
GeorgeDishman
#463
Mar24-13, 04:46 AM
P: 255
Quote Quote by RayYates View Post
As for Black Holes, since time and space are the same fabric and space seems to have been compressed out of existence, time must also have stopped, (or nearly so). In my mind I see this as a point/line extending out away from the balloon into infinity (the future). So in a way, its not part of the balloon but still attached at a single point.
If "time stops" at the event horizon, the radius would then be fixed, so it's more like a thread attached to the inside of the balloon with the other end fixed at the centre. Once it goes taut, the rest of the balloon expands as usual but that small patch gets left behind.

Like any analogy, you can only take it so far.
RayYates
#464
Mar25-13, 08:52 AM
P: 47
Quote Quote by GeorgeDishman View Post
If "time stops" at the event horizon, the radius would then be fixed, so it's more like a thread attached to the inside of the balloon with the other end fixed at the centre. Once it goes taut, the rest of the balloon expands as usual but that small patch gets left behind.

Like any analogy, you can only take it so far.
Good "point". (pun intended). I like visual image created by the thread analogy. The beginning point becomes fixed in space-time and the balloon keeps expanding into the future dragging the event horizon with it. That's a very helpful way to imagine it.
Naty1
#465
Apr23-13, 02:37 PM
P: 5,632
Jorrie posts:

Actually, there is a way in which the balloon analogy can make cosmic particle momentum decay intuitive. Simply consider a massive, frictionless particle that moves along the surface of the spherical balloon as a Kepler orbit around the center of a balloon. This particle must conserve angular momentum relative to the center of the balloon, i.e.
s:

Very nice!!. And here I thought we exhausted all the 'analogies' with Phinds BALLOON ANALOGY last year. Phinds...you should Add this and Marcus' prior post to your discussion!!


//////////////////


Quote by RayYates
As for Black Holes, since time and space are the same fabric and space seems to have been compressed out of existence, time must also have stopped, (or nearly so). In my mind I see this as a point/line extending out away from the balloon into infinity (the future). So in a way, its not part of the balloon but still attached at a single point.
Georgedishman:
If "time stops" at the event horizon, the radius would then be fixed..
I'm pretty sure there is some misunderstanding here:

The 'singularity', not the horizon, is believed to be a point in time where space has been compressed out of existence.

'time stopping' at the horizon is a local, coordinate effect. Only for an accelerating not an inertial [free falling] observer.

The event horizon is a global construct....

Illustrations :


From Kip Thorne in BLACK HOLES AND TIME WARPS
Finkelstein’s Reference Frame

when the star forms a black hole:
Finkelstein's reference frame was large enough to describe the star's implosion ...simultaneously from the viewpoint of far away static observers and from the viewpoint of observers who ride inward with the imploding star. The resulting description reconciled...the freezing of the implosion as observed from far away with (in contrast to) the continued implosion as observed from the stars surface....an imploding star really does shrink through the critical circumference without hesitation....That it appears to freeze as seen from far away is an illusion....General relativity insists that the star's matter will be crunched out of existence in the singularity at the center of the black...
Kruskal–Szekeres coordinates
http://en.wikipedia.org/wiki/Kruskal_coordinates

These coordinates have the advantage that they cover the entire spacetime manifold of the maximally extended Schwarzschild solution and are well-behaved everywhere outside the physical singularity…..The location of the event horizon (r = 2GM) in these coordinates is given by

So, a light cone drawn in a Kruskal-Szekeres diagram will look just the same as a light cone in a Minkowski diagram in special relativity.
'well behaved' here means using Kruskal-Szekeres coordinates instead of Schwarszchild time does not 'stop' at the horizon.

for further discussion, a separate thread would be appropriate.
GeorgeDishman
#466
Apr23-13, 04:11 PM
P: 255
Good points, and that is why I said you could only take it so far. Can you create an exact correspondence between coordinates on the balloon surface any any of those you listed?
Naty1
#467
Apr23-13, 04:50 PM
P: 5,632
Can you create an exact correspondence between coordinates on the balloon surface any any of those you listed?
Good question, but above my paygrade for now...

Oddly that issue did not come up in a very long thread started by phinds "Balloon Analogy'....

But the endpoints [coordinates] of paths seems to not be the only issue:

In another discussion I tried unsuccessfully to sort out the idea that the Einstein Field Equations, used in cosmology, deal with geodesics in 4D spacetime. So what does a geodesic of 4D spacetime look like in 3D space? And what does that look like on a 2D balloon surface??
marcus
#468
May16-13, 11:15 PM
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Before I lose track of the links, I'll try to get something together about current measurments of the spatial mean curvature.
Page 40 of the relevant Planck report ( http://arxiv.org/abs/1303.5076 )
==quote==
With Planck we detect gravitational lensing at about 26σ through the 4-point function (Sect. 5.1 and PlanckCollaborationXVII 2013). This strong detection of gravitational lensing allows us to constrain the curvature to percent level precision using observations of the CMB alone:

100ΩK= −4.2+4.3-4.8 (95%; Planck+WP+highL);
100ΩK= −1.0+1.8 -1.9 (95%; Planck+lensing + WP+highL)

These constraints are improved substantially by the addition of BAO data. We then find

100ΩK = −0.05+0.65-0.66 (95%; Planck+WP+highL+BAO)
100ΩK = −0.10+0.62-0.65 (95%;Planck+lensing+WP+highL+BAO)
==endquote==
Here's an earlier post on the topic of mean spatial curvature:
Quote Quote by marcus View Post
I don't think I posted anything about the WMAP9 report (Hinshaw et al.) what it said about Ωk . *I'll get a link...
==quote==
http://arxiv.org/abs/1212.5226
Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Parameter Results
G. Hinshaw, D. Larson, E. Komatsu, D. N. Spergel, C. L. Bennett, J. Dunkley, M. R. Nolta, M. Halpern, R. S. Hill, N. Odegard, L. Page, K. M. Smith, J. L. Weiland, B. Gold, N. Jarosik, A. Kogut, M. Limon, S. S. Meyer, G. S. Tucker, E. Wollack, E. L. Wright
(Submitted on 20 Dec 2012 (v1), last revised 30 Jan 2013 (this version, v2))

On page 19 you see:
Ωk= −0.0027+0.0039-0.0038 * * this is using pretty much all the major data sets: * * * WMAP +eCMB+BAO+H
eCMB or "externalCMB" includes SPT but not the LATEST SPT. But that is details.
When you translate their plus/minus stuff it leads to a confidence interval of
-0.0065 < Ωk < 0.0012

So that is lopsided on the negative Ωk side, which means FINITE but it also has some zero and positive territory which means spatial INFINITE. *Thats how several recent major reports have been going. *You can't exclude spatial infinite, at this point.

On page 20 they also have a figure −0.0065 0.0040 which I don't take as seriously but which ostensibly is based on even more data namely *WMAP +eCMB+BAO+H + SNe.
That would correspond to a purely negative interval:
-0.0105 < Ωk < -0.0025
That would exclude the spatial infinite case, at whatever the confidence level is. But this ball is still up in the air.
==endquote==
And here was an earlier post that mentioned the curvature measurement by the SPT:


Quote Quote by marcus View Post
The South Pole Telescope (SPT) has given us new narrowed-down ranges for the cosmological parameters...
http://arxiv.org/pdf/1210.7231v1.pdf
==quote==
Perhaps the most remarkable thing is the tilt towards positive overall curvature, corresponding to a negative value of Ωk

For that, see equation (21) on page 14
Ωk =−0.00590.0040.
Basically they are saying that with high probability you are looking at a spatial finite slight positive curvature. The flattest it could be IOW is 0.0019, with
Ωtotal = 1.0019
And a radius of curvature 14/sqrt(.0019) ≈ 320 billion LY.
Plus they are saying Omega total COULD be as high as 1.0099 which would mean
radius of curvature 14/sqrt(.0099) ≈ 140 billion LY.
==endquote==


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