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## Differences between GR & SR are frame-dependent

 Quote by jonmtkisco Why does the fact that the distance between the dust particles is increasing cause the peculiar velocity of the test particle to decelerate? My understanding is that peculiar velocity decays at the rate of 1/a. This tells me that gravitational deceleration of the expansion causes peculiar velocity to decay more slowly (as a function of time) than if the expansion rate were decelerating less, or not at all. Jon
Indeed it does, and on the other hand in a Lambda dominated Universe, peculiar velocities decay very quickly. The most straightforward way to see why is to look at the maths, the details are presented in this paper.

In terms of a handwaving explanation, consider first the empty universe and the tethered galaxy experiment. We set up a distant galaxy such that it has a peculiar velocity towards us that exactly balances the recession velocity away. We then let it go and see what happens.

In the empty universe described in SR co-ordinates the particle simply has no motion with respect to our chosen origin. As per the Milne model, we can postulate massless co-moving particles that have initial velocities proportional to their distance from the origin, this defines the FRW like co-moving co-ordinates. What we see is as time goes by, particles that have slower and slower recession speeds pass our test particle. Since peculiar velocity is defined as the velocity relative to local objects in the Hubble flow the peculiar velocity decays, even though the origin and the test particle have no relative motion at any point.

Now consider what happens when we add matter. It doesn't matter if the universe is flat, closed or open but we will restrict the analysis to an expanding but decelerating epoch (not a collapsing phase of a closed universe). In this case what we see is that the presence of matter will cause all velocities between all particles to decelerate. Relative to the empty universe then, the co-ordinate defining co-moving particles will start to move more slowly past our test particle, i.e. a co-moving particle midway between the origin and the tethered galaxy initially will take longer to move past the tethered galaxy in this decelerating universe than it did in the coasting universe. All of this means that the particles velocity relative to the local hubble flow remains greater for longer when there is matter in the universe.

In the case of Lambda, the reverse occurs, the co-ordinate defining particles get pushed out more quickly, hence the peculiar velocity decays rapidly.

This is just hand waving though, it's much clearer to go through the maths yourself, and see what terms would change in what way by the addition of matter.

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 non-intuitive. It is better just to say that peculiar velocity decays in fixed non-moving (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.

 Quote by Wallace Now consider what happens when we add matter. ... All of this means that the particles velocity relative to the local hubble flow remains greater for longer when there is matter in the universe.
So you agree with me that the only role of matter is to decelerate the Hubble recessional flow in an expanding universe with Lambda = 0. Adding matter to the universe seemingly does not affect the decay of peculiar velocities at all in fixed coordinates. Therefore, regardless of the matter density, matter cannot be the cause of the decay of peculiar velocity. Clearly then, "the expansion of space itself" is the sole cause of the decay of peculiar velocity. Which was my original point.

Jon

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 Quote by jonmtkisco So you agree with me that the only role of matter is to decelerate the Hubble recessional flow in an expanding universe with Lambda = 0. Adding matter to the universe seemingly does not affect the decay of peculiar velocities at all in fixed coordinates. Therefore, regardless of the matter density, matter cannot be the cause of the decay of peculiar velocity. Clearly then, "the expansion of space itself" is the sole cause of the decay of peculiar velocity. Which was my original point. Jon
Peculiar velocities don't exist in Minkowski like co-ordinates, so you cannot sensibly talk about 'the decay of peculiar velocities at all in fixed coordinates'. I've explained why peculiar velocities, as defined in co-moving co-ordinates (which is the only co-ordinates that are defined in) decay in general, and why the addition of matter changes the rate of decay. In an empty universe peculiar velocities still decay, so it is clear that there is no 'expansion of space' causing this decay, it is merely a property of a co-ordinate system. The addition of matter (or any other form of energy) changes the rate of decay, showing that it is the gravitational effect of that energy that causes any difference in the rate of decay. At no point is any co-ordinate independent 'expansion of space' required to explain this.

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 Quote by jonmtkisco Hi Chronos, I assume that your typically cryptic comment is aimed at my cosmological constant example specifically. I'm willing to go along with Wallace's response to that example, although I think it's a bit of a fine line. The cosmological constant seems to be an inherent attribute which is permanently attached to, and inseparable from, each tiny "unit" of space, so treating the energy content of space as something separate from space itself could not unreasonably be viewed as arbitrary. It's like treating the gravitational energy of matter as something separate from the matter itself. Nevertheless, I'll accept Wallace's answer, because energy is energy, regardless of what it is associated with. So here's a different question along the same lines: The peculiar velocity of a massless test particle will inherently decay over time as it travels through an expanding universe with no cosmological constant. Any gravitational sources are too distant to have any significant influence. In a comoving frame, the test particle's movement subjects it to an ongoing succession of Lorentz transformations. Is it wrong to describe this as an example of the expansion of space itself changing the motion of a test particle? Jon
Apologies for being too brief, jon. Your test particle is unphysical - i.e., what would its peculiar velocity decay with respect to?

 Quote by Chronos Your test particle is unphysical - i.e., what would its peculiar velocity decay with respect to?
Hi Chronos,

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 $$r_{p}$$ used in the Barnes & Frances paper that Wallace linked in:

 ...we will use proper distance $$r_{p}$$, which is defined as being the radial $$\left( d \theta = d \phi = 0 \right)$$ spacetime interval (ds) along a hypersurface of constant cosmic time (dt = 0). The RW metric then gives the proper distance between the origin $$\left( \chi = 0 \right)$$ and $$\chi$$ at time t to be: $$r_{p} \left( t \right) = R \left( t \right) \chi \left( t \right)$$
Applying this metric in a matter dominated Lambda=0 universe, the change in proper distance as a function of time demonstrates a decaying proper velocity over time. This is illustrated in the lower center panel of Barnes & Frances Figure 1, with the (somewhat subtle) curve of the blue line signifying decaying proper velocity.

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 $$\Omega$$ scenario as in the $$<< \Omega$$ 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

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 Quote by jonmtkisco Proper velocity decays at the same rate in the $$\Omega$$ scenario as in the $$<< \Omega$$ 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.
The proper velocities of particles, whether they be co-moving initially or not, will most certainly decay at different rates in the case of $$\Omega =1$$ and $$\Omega << 1$$. Adding more or less matter does in fact change the rate of decay of all velocities. The maths for this is all in the link I provided, though not directly, but let me know if you need more details. The exact equations aren't there, since that paper treated the equation of state as a free parameter and assumed flatness. You could re-do the analysis for a matter only universe with varying matter content and the results would show that the only thing the decay rates depend on is the matter density.

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 $$0 < w < -1/3$$.

 Quote by Wallace Adding more or less matter does in fact change the rate of decay of all velocities. ... 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 $$0 < w < -1/3$$.
By Jove, you're correct Wallace. The second bottom panel from the left in Barnes & Francis Fig 1 shows the w = -1/3 case. The particle travels a total proper distance of about 1.3 over the total time interval. Compare that to the middle bottom panel, in which the particle in a w = 0 case travels a total proper distance of about 4.4 over the same time interval.

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

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 Quote by jonmtkisco 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.
I have no idea how to make sense of this paragraph? We agree that the amount of matter in the Universe is the only thing that motions (regardless of how those motions are termed; proper, peculiar, co-moving...) depend on. In one co-ordinate system we can talk about expanding space, and in another equally valid system there appears no such term. Hence the only co-ordinate independent property that affects motion is the amount of matter we have. I have no idea how you therefore conclude that 'the only remaining candidate is the expansion of space itself'.

I'm pretty bored with this conversation though.
 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 Einstein-de 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
 Recognitions: Science Advisor 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 co-moving co-ordinates to conformally related Minkowski like co-ordinates 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.
 OK Wallace. As I look back at the posts in this thread, it isn't clear to me where you were using the conformal co-ordinates 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 Einstein-de 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
 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 near-zero gravitational deceleration. By comparison, the other universe will exhibit significant deceleration. Note also that the test particle in column 2 shows near-zero 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 mass-energy is the only factor which affects particle motions. Jon

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 Quote by jonmtkisco 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.
Yes yes, I'd forgotten about this simple way of putting it. The key to understanding the effect of expansion is that it is the second, not the first, derivative of the scale factor that is important. In other words the fact the the Universe is expanding is not important, rather how that rate of expansion is changing that is the key.
 Hi All - can't follow all this, but maybe I can pick up some terminology. To wit: 'massless test particle"? You called it non-physical ... 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.

Hi pixchips, good questions:

 Quote by pixchips Hi All - can't follow all this, but maybe I can pick up some terminology. To wit: 'massless test particle"? You called it non-physical ... so it's a mathematical construct to test something? What exactly is it testing?
The only reason to refer to the test particle as "massless" is to make it clear that for the purposes of this exercise we're ignoring any effects caused by the active gravity of the test particle itself. It is not intended to imply that the test particle does not feel the effect of gravity from other objects. Nor is intended to imply that the test particle is relativistic, such as a photon.
 Quote by pixchips 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.
Correct.
 Quote by pixchips But you seem to say that in some coordinate systems there is no expansion ... is the coordinate system expanding??
A "proper distance" coordinate system does not expand, so observers moving apart in the Hubble flow measure each other as moving apart in these coordinates. On the other hand a "comoving" coordinate system can be thought of as expanding exactly in synch with the Hubble flow. So all comoving observers are considered to be "at rest" with respect to each other, even though the proper distance between observers increases over time due to the Hubble flow.

Hope that helps.

Jon
 Yes, thanks. The co-moving coordinate system is a new concept for me. It would perhaps be like drawing a cross hatched x-y 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 ......
 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

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