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Differences between GR & SR are framedependent 
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#19
Apr2708, 07:24 PM

P: 531

Hi Wallace,
First, thanks for the link to the Barnes & Francis paper. I've read several of their papers and this one is excellent, nicely explanatory, as were the others. I don't think their point by point dissection of terminology is at all pedantic; it is only through careful exposition of all of the terminology that the fog is lifted. Their conclusion doesn't surprise me: "We contend that the problem is not that expanding space has mislead us, but that describing the decay of v(pec) as joining the Hubble flow is a misnomer." I agree that the notion of "rejoining the Hubble flow" is rather unhelpful and nonintuitive. It is better just to say that peculiar velocity decays in fixed nonmoving (proper distance) coordinates in an expanding universe, period. Converting to comoving coordinates only complicates and confuses this particular analysis. Peculiar velocity may or may not decay asymtoptically close to zero (in fixed coordinates) as time approaches infinity. I think the converse terminology about gravitationally bound objects "breaking away from the Hubble flow" also is a misnomer for exactly the same reason. Jon 


#20
Apr2708, 08:41 PM

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P: 1,253




#21
Apr2708, 11:28 PM

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PF Gold
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#22
Apr2808, 12:42 AM

P: 531

I didn't mean to suggest that in this example the universe is entirely empty. As I further explained to Wallace, "Let's assume that the mass of the universe is distributed homogeneously as dust. Lambda is zero, and the matter is at critical density." As I also tried to convey (apparently I misstated it) in my most recent post, I want to adopt the same definition of proper distance [tex]r_{p}[/tex] used in the Barnes & Frances paper that Wallace linked in: Wallace, in response to your comments, I find references to the Milne model to create more confusion than they resolve, due to the rather unique characteristics of that model. I would prefer to engage the discussion in terms of an Einstein de Sitter model: In one scenario it is very far below critical density (but not empty); in the second scenario it is at critical density. Proper velocity decays at the same rate in the [tex] \Omega[/tex] scenario as in the [tex] << \Omega[/tex] scenario. Therefore I continue to conclude that matter cannot be the cause of proper velocity decay. Adding more matter neither increases nor decreases the amount of proper velocity decay. (The only effect of matter is to decelerate the background recession velocity, which actually slows down decay in comoving coordinates.) If matter is not the cause of the decay, then what is? As far as I can see, the expansion of space is the only remaining candidate. Jon 


#23
Apr2808, 05:49 AM

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P: 1,253

I can be sure of this without doing the full calculation since for radial motion an underdense matter universe in equivalent to a Universe with overall equation of state that varies with time between [tex] 0 < w < 1/3 [/tex]. 


#24
Apr2808, 11:43 AM

P: 531

Thus, adding matter to the universe retards the decay of both the proper velocity and the comoving peculiar velocity. Which strengthens the conclusion that matter cannot be the cause of the decay. The only remaining candidate is the expansion of space itself. Jon 


#25
Apr2808, 06:46 PM

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P: 1,253

I'm pretty bored with this conversation though. 


#26
Apr2808, 08:44 PM

P: 531

Hi Wallace,
The conversation goes nowhere because you want to tie all of your justifications to the Milne model which is not a useful representation of the real universe. It's a stilted model which gets attention because apparently it is the only exact solution so far to the Einstein equation for an empty universe. If we can engage in a dialogue based on the varying density within the Einsteinde Sitter model, maybe we can shake the pressureless dust off of this conversation. There is a good discussion to be had on this subject. Jon 


#27
Apr2908, 07:06 AM

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P: 1,253

Jon, we've been discussing models with varying matter density for quite some time now. The conversation you claim would be 'a good discussion to be had' is in fact the one we've been having. I really have no idea how you can suggest I'm basing everything 'on the Milne model'. Remember that in any general matter model you can transform from comoving coordinates to conformally related Minkowski like coordinates as shown in the Chodorowski paper I linked to. This is not the Milne model, this is any FRW model with matter. That's what we've been talking about for a while now.



#28
Apr2908, 11:58 AM

P: 531

OK Wallace. As I look back at the posts in this thread, it isn't clear to me where you were using the conformal coordinates instead of the Milne model, and how that related specifically to the math and diagrams in the Francis and Barnes paper.
Well, I appreciate your patience in discussing this as much as we did. Now I'm going to try to draw a politically neutral conclusion from the parts of this discussion I understand: In an expanding Einsteinde Sitter universe with Lambda=0, both proper velocity and comoving peculiar velocity decay at rates which are inversely related to the matter density. The presence of matter affects velocity decays, but cannot be attributed as their cause. Jon 


#29
May308, 09:16 PM

P: 531

Well, after giving this some further consideration, I now believe that my politically neutral conclusion is wrong. There are a lot of moving parts in this analysis, and it's tough to keep them all straight. Sigh...
In Figure 1 in the linked Barnes & Francis paper, column 2 (w = 1/3 or matter density << Omega) in the lower panel indeed shows that the test particle travels a shorter proper distance over the time period than the test particle in column 3 (w = 0, matter density = Omega) in the lower panel travels. However, in my last post I was incorrect in attributing this shorter travel distance in column 2 to increased proper velocity decay. The correct reason is that the initial conditions of the two columns are different. In fact, the initial proper velocity is lower in column 2 than in column 3. The reason for this is that a universe with density << Omega (Lambda = 0) expands more slowly (at a given proper elapsed time since t=0) than a universe with density = Omega (Lambda = 0). Since the expansion velocity is lower in the underdense universe, the "tethered" test particle will of course start this exercise with a lower peculiar velocity (equal to the lower expansion rate.) If the matter density in the underdense universe is near zero, then its expansion rate will exhibit nearzero gravitational deceleration. By comparison, the other universe will exhibit significant deceleration. Note also that the test particle in column 2 shows nearzero decay in proper velocity (i.e., the blue line is straight, not curved up as in columns 3 and 4.) My interpretation of the relationship between columns 2, 3 and 4 is that the decay in the proper velocity of the test particle is due entirely to the deceleration factor of the background expansion rate, and not at all to the velocity of the background expansion rate. So the rate of decay in proper velocity is positively correlated with increasing matter density, rather than inversely correlated as I had previously interpreted. For this reason, it makes sense to conclude that the presence of matter is the sole cause of the decay in proper velocity. It seems to me that the mostly likely explanation is that an expanding but decelerating background dust field exerts a decelerational gravitational pull on a test particle passing peculiarly through it. With this change in conclusion, this example no longer demonstrates that expanding space affects the motion of a test particle. Chalk up another example for Wallace's explanation that the presence of massenergy is the only factor which affects particle motions. Jon 


#30
May608, 10:46 AM

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P: 1,253




#31
May1908, 11:57 PM

P: 50

Hi All  can't follow all this, but maybe I can pick up some terminology. To wit: 'massless test particle"? You called it nonphysical ... so it's a mathematical construct to test something? What exactly is it testing? (My guess  massive particles don't follow the Hubble flow because of their inertia, so a massless particle would just 'go with the flow'  can't be a real massless particle since it would have to tool around at c all the time.) Also: "peculiar velocity" seems to be the velocity relative to the 'Hubble Flow', which seems to be the 'average' motion of space due to expansion. But you seem to say that in some coordinate systems there is no expansion ... is the coordinate system expanding?? Okay, that's enough for now .... sorry to intrude, but sometimes the kids listen in when the adults are talking, and they have questions.



#32
May2208, 01:12 AM

P: 531

Hi pixchips, good questions:
Hope that helps. Jon 


#33
May2208, 02:36 AM

P: 50

Yes, thanks. The comoving coordinate system is a new concept for me. It would perhaps be like drawing a cross hatched xy coordinate (latitude/longitude) system on a balloon and then blowing up the balloon? Things stuck to the balloon, at rest relative to its surface, separate as the balloon expands relative to a proper distance frame but don't move relative to the cross hatch pattern I drew on the balloon. The cross hatch is the comoving coordinate system? So this is a mathematical convenience to help talk about things relative to the Hubble Flow? In this coordinate system geometric relations are fixed, but scale is constantly changing? Hmmm ... I need to grok this ......
Simple example: two comoving charges would have a force between them that goes as 1/'proper distance'^2. To change to the comoving coordinate system, I would need to change Maxwell's equations ... but how can I do that? If they are comoving (I don't know what's got them glued in place, but for sake of argument ....), then in the comoving coordinate system the distance doesn't change. So my law for attraction of charges would then be dependent on distance and time .... I do not grok this ...... 


#34
May2208, 02:29 PM

P: 531

Hi pixchips,
Your description of comoving coordinates is basically right. The "dots on a balloon" model is one of the standard analogies used to describe this concept, although it's explanatory power is subject to some important limitations. If you haven't, check out Wikipedia on Metric expansion of space. I'm not the best one to give a technical answer to your questions about Maxwell's equations. But I think that conceptually, Maxwell's equations will need to be recalculated at each instant in time, to address the fact that two massive objects are moving apart from each other. In everyday terms, the Hubble flow is so tiny at the distances over which electromagnetism is significant that it makes no significant difference. And if two charged objects are close enough to be electromagnetically (or gravitationally) bound together, they don't move apart. Jon 


#35
May2208, 11:52 PM

P: 50

Thanks ... yeah, I knew about the balloon, but nobody drew a coordinate system on it and called it the comoving coordinate system.. Anyway, just hooked into a couple much deeper references (for me anyway) and I'm working through those. Thanks for your attention.



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