The balloon analogy (please critique)

[phinds]

I want to share an 'advanced version' of the balloon model...but one that has it's own issues. I think this illustration would be a good follow on for phinds to consder adding as a link to his balloon site.

[If you click on the illustration it will blow it up...necessry if you are old like me! ]


http://en.wikipedia.org/wiki/Metric_expansion#Understanding_the_expansion_of_Universe

The updated version already has this link

 

[Ich]

I don't see the first part right off since I thought we can pick space and time coordinates arbitrarily. And we are not moving with the expansion as we measure, but we can measure I guess at a fixed time...maybe that's the implication..

Yes, by fixing cosmological time you cut a three space out of 4D spacetime. It's similar to cutting the balloon surface out of 3D space.
In the balloon it is clear that the geodesics of the subspace - great circles - are not geodesics of the embedding space, which would be straight lines.
Same with FRW space, proper distance is measured along geodesics of the subspace, which are not geodesics of spacetime.

 

[GeorgeDishman]

No offense, but IMHO you're saying that because you're not aware of said subtleties.

By all means start another thread then, or feel free to message me if you feel it more appropriate, I am always keen to improve my understanding.

You forget the this ruler is made out of infinitely many segments which all have relative velocity wrt each other.

I didn't forget, that is why I included the highlighted qualifier:

Cosmological distance is defined as the sum of a set of rulers which happen to be laid exactly end to end at a particular cosmological time which directly corresponds to the ruler on a sheet of paper. ...​

Which is not exactly what you have in your household.

At that particular moment, it corresponds exactly IMHO, but please correct me if I have missed something.

 

[Naty1]

From post #49

Cosmological distance is defined as the sum of a set of rulers which happen to be laid exactly end to end at a particular cosmological time which directly corresponds to the ruler on a sheet of paper...

Ich

You forget the this ruler is made out of infinitely many segments which all have relative velocity wrt each other.

How can they have relative velocity if the cosmological moment in time is fixed?



Ich:

...the so-called recession "velocity" is rather a rapidity, which goes quite naturally beyond c. This is important if one wishes to discuss "superluminal" recession "velocities...

Of course! great point! Therefore, I shall continue to keep reading your stuff, Ich!

 

[Naty1]

pHinds
sorry for all this, discard what you like..my last comments!

Regarding your Balloon Analogy website
I like it! Well done...It should get put in FAQ in these forums


[1] Should the balloon analogy be linked to the FLRW model?? I'm unsure.

Ich seems to think in a post here it is. I think you should mention there are not precise measures of distance and time in cosmology...we use conventions to allow us to make agreed upon measures, standard comparisons. But overall, the arbitray split between space and time of different observers leads to 'ambiguity' [using a word in the wiki reference].


Under "third local effect" :

The pennies don't change size (gravitationally bound systems don't expand and nothing inside of them expands), they just get farther apart and none of them are at the center.

Correct me, somebody, if I misinterpreted another thread discussion, but I thought that the FLRW model [homogeneous, isotropic] did NOT apply at galactic distances...too much lumpiness. In addition I thought nobody knows how to solve the EFE for representative galactic conditions...how to include the lumpiness in other words. So should we instead say something like 'gravitationally bound systems and things inside them are not thought to expand [or are generally not considered to exapnd] but we have no exact solution for such conditions. I'm not sure.

[3] In your description, Second Size shape:

Forget that the surface of the balloon is curved. That's NOT intended to be representative of the actual universe. It is actually more reasonable to think of a flat sheet of rubber that is being stretched equally in all directions.

Last sentence: Should this be qualified to space versus spacetime. Or say that curvature in time is not represented in the balloon analogy. We believe the universe is pretty flat spacewise, right? Is it time that is mostly curving on cosmological scales...or not??


[4] Cosmological Time: How do we say in a sentence or two, and should we bother here, that

Cosmological time is the elapsed time since the Big Bang according to the clock of an observer comoving with the CMBR ...[we use the cosmological time parameter of comoving coordinates because it's convenient mathematically. There are other time measures that could also be used.] In the Wikipedia link above, cosmological time, the 'age of the universe', is the like the time of light transit along the red curve, about 13B years, not the transit time along today's orange curve distance which is about 28B years.



[4] Under OTHER NOTES

How about a few sentences like this :

"Sending a light signal from one penny [galaxy] to another will take longer than if the pennies were stationary with respect to each other because the distances between them are increasing. [DUH!] Because the actual rate of expansion is not constant over all of cosmological time, the Hubble 'constant' varies over time since the big bang, and the actual transit time between pennies is different today than it was at earlier times. The current expansion of the universe proceeds in all directions as determined by the Hubble constant today, but it is a 'constant' in all directions of space not over time.

 

[phinds]

pHinds
sorry for all this, discard what you like..my last comments!

Regarding your Balloon Analogy website
I like it! Well done...It should get put in FAQ in these forums


[1] Should the balloon analogy be linked to the FLRW model?? I'm unsure.

Ich seems to think in a post here it is. I think you should mention there are not precise measures of distance and time in cosmology...we use conventions to allow us to make agreed upon measures, standard comparisons. But overall, the arbitray split between space and time of different observers leads to 'ambiguity' [using a word in the wiki reference].


Under "third local effect" :



Correct me, somebody, if I misinterpreted another thread discussion, but I thought that the FLRW model [homogeneous, isotropic] did NOT apply at galactic distances...too much lumpiness. In addition I thought nobody knows how to solve the EFE for representative galactic conditions...how to include the lumpiness in other words. So should we instead say something like 'gravitationally bound systems and things inside them are not thought to expand [or are generally not considered to exapnd] but we have no exact solution for such conditions. I'm not sure.

[3] In your description, Second Size shape:



Last sentence: Should this be qualified to space versus spacetime. Or say that curvature in time is not represented in the balloon analogy. We believe the universe is pretty flat spacewise, right? Is it time that is mostly curving on cosmological scales...or not??


[4] Cosmological Time: How do we say in a sentence or two, and should we bother here, that

Cosmological time is the elapsed time since the Big Bang according to the clock of an observer comoving with the CMBR ...[we use the cosmological time parameter of comoving coordinates because it's convenient mathematically. There are other time measures that could also be used.] In the Wikipedia link above, cosmological time, the 'age of the universe', is the like the time of light transit along the red curve, about 13B years, not the transit time along today's orange curve distance which is about 28B years.



[4] Under OTHER NOTES

How about a few sentences like this :

"Sending a light signal from one penny [galaxy] to another will take longer than if the pennies were stationary with respect to each other because the distances between them are increasing. [DUH!] Because the actual rate of expansion is not constant over all of cosmological time, the Hubble 'constant' varies over time since the big bang, and the actual transit time between pennies is different today than it was at earlier times. The current expansion of the universe proceeds in all directions as determined by the Hubble constant today, but it is a 'constant' in all directions of space not over time.

Good points. I've added a note at the bottom of the page that encompasses much of this (but without actually getting INTO any of it), for the reason stated in the note:

NOTE: measures of distance and time in cosmology, as well as the shape/extent of the universe and the fact that "space" is really "space-time", are all very complex topics, and my simplistic ways of talking about them on this page are just that ... simplistic. My point here was to produce a fairly modest, but correct (with some simplifcations) analysis of the balloon analogy without, as I noted at the beginning, writing a text on cosmology.

EDIT: by the way, thanks again to all for the continued feedback.

Paul

 

[GeorgeDishman]

Note the orange line [present day distance] follows the purple grid curve of constant time. ... You can see from this illustration there are many other curves we could pick...and each gives a different measurement. The orange [FLRW metric distance] line is not directly observable from Earth and that is why it doesn't compare closely in my opinion with the curved surface distance of the balloon analogy.

First you need to imagine the sheet continued so the ends join up. That creates a picture like the one below with the galaxies etc.. Take the orange grid line (which is now a complete circle) and rotate it to create a sphere. That is your balloon.

You experts on all this stuff can correct me on this but I did NOT think the orange distance a curve could possibly be the 'geodesic' light would follow...since light takes a finite time to travel.

You are correct. The red line is a null geodesic which is the path that the light took to reach us from the quasar. As the page says, it crosses each grid line at 45 degrees. Any massive particle must travel more slowly hence must cross the cyan lines at less than 45 degrees.

To my way of thinking, so far, one could pick any number of curves on the balloon surface to measure penny separation distances. We would need to agree on a convention, and a great circle arc would be a natural.

If the balloon surface is uniform, distances between galaxies grow at a rate which is proportional to their separation. That is the Hubble Law and that law holds for comoving distances, the distance measured by the orange arc.

That does seem analogous to choosing a convention for a distance metric.

The balloon illustrates the FLRW metric.

Here is an issue I had not thought about before:
What about dips around pennies to illustrate local galaxy gravitational irregularities?? Maybe the idea of 'dip' is a non starter because the FLRW metric assumes homogeaneity so we skip those in our calculation.

The surface looks smooth at large scales but closer up it looks like the skin of an orange.

I dunno, but CMBR sure has to follow such dips when we measure redshift, right...but there is supposedly no expansion within galaxies, no distance increases, so no redshift, so no observational effect?

The dip extends beyond the galaxy, that's what creates gravitational lensing:

http://apod.nasa.gov/apod/ap090921.html

See also the Integrated Sachs-Wolfe Effect:

http://en.wikipedia.org/wiki/Sachs–Wolfe_effect#Integrated_Sachs.E2.80.93Wolfe_effect

 

[Naty1]

On distance analogy:
I have three [oops, four] perspectives that I am trying to sort thru regarding the appropriatness [accuracy] of the balloon analogy to FLRW metric.

Ich
Which is not exactly what you have in your household.

George:
At that particular moment, it corresponds exactly IMHO, but please correct me if I have missed something.

George, Thats my perspective, so far, as well. It's what I questioned in my post #49. I also feel like I might be missing something.

A second related point is this which I already posted:

The orange [FLRW metric distance] line is not directly observable from Earth and that is why it doesn't compare closely in my opinion with the curved surface distance of the balloon analogy.

[George noted:

The red line is a null geodesic which is the path that the light took to reach us from the quasar. As the page says, it crosses each grid line at 45 degrees.

Of course! I did see that then forgot!...That's a really nice reference especially if phinds is aiming his Site at beginners: It ties the Wiki diagram to the traditional lightcone used with Minkowski spacetime. The connection is not so obvious for those starting out! Such visual links between concepts can really cement new concepts in place.


#3: I happened to be rereading LineWeaver and Davis since I haven't in a long time and they make this interesting statement:

The microwave background radiation fills the universe and defines a universal
reference frame, analogous to the rubber of the balloon, with respect to which motion can be measured

I had this same thought earlier and forgot to post it. I consider it a useful analogy. Seeing this analogy several years ago would have made me realize that in the balloon model we are observing the 'universe' from the outside and that can't be done in the 'real world' ! We are stuck on the surface and this statement begins to define the FLRW metric distance convention.

#4: My last issue is the earlier posted point from Wallace regarding acceleration not velocity [or rapidity if your prefer] as the determining factor in separation. The balloon analogy does NOT capture that but how to explain in simple terms why is not yet clear to me...

 

[Naty1]

George:

If the balloon surface is uniform, distances between galaxies grow at a rate which is proportional to their separation. That is the Hubble Law and that law holds for comoving distances, the distance measured by the orange arc.

That's a nice observation regarding CURRENT distance measures...since the Hubble constant varies over time. It's obvious, but I did not think of it...thanks!



me:

I dunno, but CMBR sure has to follow such dips when we measure redshift, right...but there is supposedly no expansion within galaxies, no distance increases, so no redshift, so no observational effect?

George:

The dip extends beyond the galaxy, that's what creates gravitational lensing

See also the Integrated Sachs-Wolfe Effect:

http://en.wikipedia.org/wiki/Sachs%E...93Wolfe_effect

I read that for the first time a few weeks ago and never got around to posting my basic question about it. Wikie says

Accelerated expansion due to dark energy causes even strong large-scale potential wells (superclusters) and hills (voids) to decay over the time it takes a photon to travel through them. A photon gets a kick of energy going into a potential well (a supercluster), and it keeps some of that energy after it exits, after the well has been stretched out and shallowed. Similarly, a photon has to expend energy entering a supervoid, but will not get all of it back upon exiting the slightly squashed potential hill.

What I wondered in the article is whether such potential well 'detours' of CMBR photons require or deserve any correction in CMBR observations??

 

[GeorgeDishman]

#4: My last issue is the earlier posted point from Wallace regarding acceleration not velocity [or rapidity if your prefer] as the determining factor in separation. The balloon analogy does NOT capture that but how to explain in simple terms why is not yet clear to me...

Think of Lineweaver's ants and aphids. Suppose an ant gets tired of walking ever farther and ties a rope to an aphid. By the time he gets home, the rope is slipping through his hand if he stands still. He puts on roller skates and grabs the rope. Now he is moving across the rubber. If the inflation rate of the balloon falls, the rope goes slack and he coasts back to the aphids, no effort involved ;-)

That is however very different to saying the rate of increase of distance between his home and the aphids only depends on the acceleration.

 

[GeorgeDishman]

If the balloon surface is uniform, distances between galaxies grow at a rate which is proportional to their separation. That is the Hubble Law and that law holds for comoving distances, the distance measured by the orange arc.

That's a nice observation regarding CURRENT distance measures...since the Hubble constant varies over time. It's obvious, but I did not think of it...thanks!

That's not quite the point. For any given cosmological time, the Hubble Law is a linear relationship, rate of recession equals the constant times a distance. That is also true of separations measured on the surface of the balloon. If you use other distance measures (luminosity distance, angular size distance, etc.) for the analogy, the relationship will not be linear so it would no longer match the balloon.

What I wondered in the article is whether such potential well 'detours' of CMBR photons require or deserve any correction in CMBR observations??

The EM emissions from the galaxies themselves are generally greater so "foreground features" have to be removed. However, we can use the effect to learn about the galaxies since the CMBR is so well defined.

 

[Ich]

#4: My last issue is the earlier posted point from Wallace regarding acceleration not velocity [or rapidity if your prefer] as the determining factor in separation. The balloon analogy does NOT capture that but how to explain in simple terms why is not yet clear to me...

What he meant by "velocity" is \dot a, the time derivative of the scale factor. It corresponds to the radial velocity of the balloon surface. (It is its proper velocity rather, not bounded by c therefore.)
The acceleration is \ddot a, here the proper radial acceleration of the balloon surface.

Now if you put two dots at rest wrt each other on the surface (i.e. not comoving), their relative acceleration is proportional to \ddot a, not \dot a. That holds in FRW coordinates as well as in the analogy.

I'll open another thread for the distance definition subtleties, that doesn't belong here.

 

[Naty1]

Ich, George,,,thanks for the feedback...appreciate it...

will reread your explanations tomorrow and be back then...

But not until I walk my Yorkies...after all, this is JUST science...!

Idea of a separate discussion on distance is good... look forward to that!

 

[Naty1]

George's Ned Wright link posted above did not 'click' for me after an initial reading so I was doing some background reading and came across this Wikipedia discussion which seems to support my own incorrect interpretation... not what George claimed for Wright...but in all honestly, Wright's explanation link and this one below are not really clear to me yet:

http://en.wikipedia.org/wiki/Comoving_distance#Uses_of_the_proper_distance

...It is important to the definition of both comoving distance and proper distance in the cosmological sense (as opposed to proper length in special relativity) that all observers have the same cosmological age. For instance, if one measured the distance along a straight line or spacelike geodesic between the two points, observers situated between the two points would have different cosmological ages when the geodesic path crossed their own world lines, so in calculating the distance along this geodesic one would not be correctly measuring comoving distance or cosmological proper distance. Comoving and proper distances are not the same concept of distance as the concept of distance in special relativity. This can be seen by considering the hypothetical case of a universe empty of mass, where both sorts of distance can be measured. When the density of mass in the FLRW metric is set to zero (an empty 'Milne universe'), then the cosmological coordinate system used to write this metric becomes a non-inertial coordinate system in the flat Minkowski spacetime of special relativity, one where surfaces of constant time-coordinate appear as hyperbolas when drawn in a Minkowski diagram from the perspective of an inertial frame of reference.[4] In this case, for two events which are simultaneous according the cosmological time coordinate, the value of the cosmological proper distance is not equal to the value of the proper length between these same events,(Wright) which would just be the distance along a straight line between the events in a Minkowski diagram (and a straight line is a geodesic in flat Minkowski spacetime), or the coordinate distance between the events in the inertial frame where they are simultaneous...[/QUOTE

Maybe this is better saved for a subsequent discussion on distance...I did want to post it for future reference.

I assume I am the one that is 'mixed up' and will continue background reading...

 

[Ich]

Sorry for the delay, I'll start the other thread tomorrow (I hope). Again, it will go along the line of Ned Wright's arguments.
For the time being: a spacelike geodesic is not the same as a geodesic of space. The former is a geodesic of spacetime which is, well, spacelike. The latter is a curve of extremal distance in some subspace of spacetime, which is necessarily spacelike but not necessarily also a geodesic of spacetime.