Hi Jorrie,

Good post as usual. I see you invited me to respond in the last sentence. The 'drawings' of this post make it much clearer to 'view' the situation as compared to the previous discussion about tethering. The first figure is clear as far as it goes. In the second one, D_proper is shown as flat. If I assume that the 'reference' button is on the center line of the hyper-dimension, then the two buttons behave the same as if they are still tethered? That is not shown in the first figure, so it's not as clear. Your text seems to say that: "the button's peculiar velocity (across the balloon surface) towards the origin is canceled by the balloon's expansion that is trying to carry it away from the origin."

This is not what I would have expected. I would have expected it to be somewhere between the red line and the blue line. I take it that momentum is keeping it there? The previous thread is used as an explanation. Correct me if wrong. There the momentum is from an object at the same point as we are in space. Here the 'buttons' are far apart on the balloon's surface. Aside from that, I'm not sure the analogy works for momentum anyway. I don't think anybody knows what causes it, so expansion of space may reduce it by a different factor.

I'm also confused by the v_proper line in figure 2. It looks like you are defining it to be zero, which would mean that the buttons are still tethered?!? Otherwise it makes sense. Straighten me out if you can.

-S

Hi S, no you are the one that never shy away from asking, making good points as you go. I was talking to others...

"If I assume that the 'reference' button is on the center line of the hyper-dimension, then the two buttons behave the same as if they are still tethered? That is not shown in the first figure, so it's not as clear."

In this case of uniform expansion (no deceleration or acceleration), yes, the moving button will behave as if it is still tethered, although it was cut loose. Remember that I've attached the string to the origin, so the 'origin button' never moves space-wise. Remember also that the red button was given a peculiar velocity by the string that was precisely equal to the Hubble velocity at that distance.

"I take it that momentum is keeping it there? The previous thread is used as an explanation. Correct me if wrong. There the momentum is from an object at the same point as we are in space. Here the 'buttons' are far apart on the balloon's surface."

Remember, peculiar velocity is relative to the local balloon surface, wherever the button happens to be. In order for angular momentum relative to the balloon's center to remain constant, the peculiar velocity decays as per the green curve. At the same time Hubble velocity decays in identical fashion, so the balance is maintained and the red V_proper curve stays zero for ever.

These are crucial concepts; that's why the simplest case (uniform expansion) must be fully understood before we can carry on.

-J

Hi Jorrie,

"It is a rather complex issue with significant pedagogical value"

I'm not sure what you had in mind here, but I think this could be a useful teaching (pedagogy) tool. It occurred to me that this explains (the theory of) why galaxy diameters do not expand with the universe expansion. You have implied in past threads that this analogy works even in an infinite universe. The data suggests that we have one. Let's see if we can prove that the analogy works for an infinite universe.

Here's the mathematical challenge if you can accept it. So angular momentum is responsible in the balloon analogy for keeping the buttons (stars within galaxies in this case) from moving apart as the universe expands. Now let's represent the universe as a cylinder with constant diameter which expands by getting longer (easy to visualize). Let's say it is 1Gly long to start and expands to 2Gly. The buttons could be 0.1Gly apart to start, then un-tethered. Now they have 'linear' momentum toward each other with respect to expanding space. Do they still maintain the same proper distance after 2Gly?

-S

Hi S. Your "It occurred to me that this explains (the theory of) why galaxy diameters do not expand with the universe expansion" sounds a bit off the mark to me.

It is just like the button diameter that does not increase (expand), being held constant by molecular forces. The stars in a galaxy are like the molecules of the buttons, with molecular forces replaced by gravity. There is a feeble 'tidal force' on galaxies in a universe with changing rate of expansion, but utterly negligible compared to gravity.

On your 'cylinder analogy'. Yes, you can represent a piece of an infinite, flat universe like that - it's more or less like taking a small section of skin of a very large balloon, where you cannot observe the curvature.

"Let's say it is 1Gly long to start and expands to 2Gly. The buttons could be 0.1Gly apart to start, then un-tethered. Now they have 'linear' momentum toward each other with respect to expanding space. Do they still maintain the same proper distance after 2Gly?"

Yes, but only in the hypothetical 'constant expansion rate' case (which represents an empty universe (0,0,0)). Perhaps I should have given the realistic case plots as well, but I thought it 'good pedagogy' to treat the simplest case first. In the realistic case, the button starts to move away from the origin virtually immediately, provided we are in an increasing expansion rate phase. Here is the graph, without much explanation (which I will do in the main article soon).

Jon and myself have discussed the "cosmic tidal force" issue in that long thread "A blast from the past" (I think). In a nutshell, accelerating expansion does weaken gravity a teeny-weeny bit on the scale of a galaxy, but it is still negligible even at those distances.

-J

Hi Jorrie,

"The stars in a galaxy are like the molecules of the buttons, with molecular forces replaced by gravity."

To me the orbiting from gravity seems the same as tethering. One object is accelerating the other "toward" it. I was not getting into whether atoms or molecules expand.

"you can represent a piece of an infinite, flat universe like that - it's more or less like taking a small section of skin of a very large balloon, where you cannot observe the curvature."

But I was asking if the same math applies to angular and linear momentum? If it does, then you have proved that the balloon analogy works on an infinite universe! If not, are you using angular momentum math on your small section of skin?

Me: Do they still maintain the same proper distance after 2Gly?"

You: "Yes, but only in the hypothetical 'constant expansion rate' case (which represents an empty universe (0,0,0))"

Wouldn't the same apply to when there is matter but vacuum energy that exactly cancels the gravitational pull?

-S

Hi S.

"To me the orbiting from gravity seems the same as tethering."

Yes, I suppose one can think like that, but since the buttons themselves are galaxies here, I still think the molecular forces are the best analogy. One can obviously think of a group of buttons as a galaxy cluster, being kept in orbit by tethers.

"But I was asking if the same math applies to angular and linear momentum?"

This is a bit tricky, because linear and angular momentum are not the same thing, so the math is not the same in general. However, the decay of peculiar velocity on the balloon and in a perfectly flat universe is exactly the same. For the hypersphere, we can work with the conservation of angular momentum relative to the center and get the law for the decay of peculiar velocity. In a perfectly flat universe, there is no such center...

However, as I've said to Roger before, for a flat universe it is accepted^[1] that one can choose any R larger than the Hubble radius and do your calculations in spherical coordinates. The results are not in the least influenced by the radius that you choose.

I would still not say that "the same math applies to angular and linear momentum?", but rather that the spherical math gives the proper answers for a spherical as well as for a flat cosmos.

You previously: "Now they have 'linear' momentum toward each other with respect to expanding space. Do they still maintain the same proper distance after 2Gly?"

Me "Yes, but only in the hypothetical 'constant expansion rate' case (which represents an empty universe (0,0,0))"

You: "Wouldn't the same apply to when there is matter but vacuum energy that exactly cancels the gravitational pull?"

No, this not generally true. Proper velocity becomes temporarily constant in such a case, but the proper distance will only remain constant if proper velocity remains zero, which is not generally the case in the 'middle ages' of the universe. My next update of the article will include the 'middle ages' epoch of expansion.

-J

[1] Profs. Peebles and Peacock both mention this in their respective cosmology text books.

I wrote: "My next update of the article will include the 'middle ages' epoch of expansion."

Done it. See OP.

-J

Hi Jorrie,

"Yes, I suppose one can think like that, but since the buttons themselves are galaxies here, I still think the molecular forces are the best analogy. One can obviously think of a group of buttons as a galaxy cluster, being kept in orbit by tethers."

I was trying to say you have a 'teaching aid' here, and it applies to more than whole galaxies. I can understand your 'tunnel vision' here because galaxies are what you are concentrating on.

Just looked at your updates above. Fig 4 is very interesting but I am also confused by it.

"The Hubble velocity at D=-0.1 Gly was a whopping –0.335c and hence the peculiar velocity of the tethered galaxy at that time was 0.335c towards the origin. But, due to the decreasing expansion rate, the Hubble velocity quickly diminished for that proper distance and hence when untethered, the galaxy started to 'fall' rapidly towards the origin. It 'fell through' the origin at some 1.5 billion years and then continued to move in the positive D direction."

I assume that the scale on the left side is only for distance, since you stated the Hubble velocity was -0.335c at -0.1 D. How can distance be negative? Do you mean that the two galaxies collided, and then by momentum kept going and separated going the opposite direction? That would be possible without many stars colliding. The real kicker here is how did the galaxies get separated in the first place if they are coming together then?

-S

Hi S.

"I assume that the scale on the left side is only for distance, since you stated the Hubble velocity was -0.335c at -0.1 D."

No, I just used a bit of 'zoom' for effect (better curves), so we can't see the whole of the scale. The velocity scale is 1:1, while the distance scale is D/10 as indicated. I will change the picture in the OP to show the whole velocity scale. Thanks for the tip

"How can distance be negative?"

On the cut through the balloon (Fig. 1), negative D is just to the left, since we are doing our sums in 1-D space here.

"Do you mean that the two galaxies collided, and then by momentum kept going and separated going the opposite direction?"

No, it assumes that they have moved cleanly through each other, without effecting each other in any way, not even gravitationally. Maybe I should have just called them buttons (or particles) and not galaxies, because I totally ignore their masses. The idea of the whole thought experiment is to show the peculiar and proper movements of particles caused by expansion in a homogeneous cosmos. Hmm... Better description required.

"The real kicker here is how did the galaxies get separated in the first place if they are coming together then?"

The thought experiment placed them apart, tethered them and then cut the tether. I can see you think gravitational interaction between two galaxies here. This is not part of the thought experiment. We may have to think about a different, massless, collisionless setup. Cosmologists use it as I did, but I agree that it may be confusing to the 'uninitiated'.

-J

Hi Jorrie,

The graph is much clearer now with the scale matching the description.

Me: "Do you mean that the two galaxies collided, and then by momentum kept going and separated going the opposite direction?

You: "No, it assumes that they have moved cleanly through each other, without effecting each other in any way, not even gravitationally."

That's basically what I said, or tried to say, when I stated that no stars would collide.

"The thought experiment placed them apart, tethered them and then cut the tether. I can see you think gravitational interaction between two galaxies here."

O.K., I did this late last night, and was confusing the thought experiment with the real universe. Still one confusion issue: It starts out with D_proper negative. I thought you always start with the reference on the origin and the other one positive. Why not this time, or am I more confused than I think?

-S

Hi again, S.

"That's basically what I said, or tried to say, when I stated that no stars would collide."

I later realized that you have thrown me a curve-ball here and I've missed it! I originally described the experiment thus:

"Pick any spot on a partially inflated cosmic balloon's surface and attach one end of a tether (string) there. Call this spot the origin of the coordinate system. At the other end of the tether, attach a button lying frictionless on the surface - the button is now 'tethered' to the origin at a distance D from it."

Only one button (galaxy), a tether and an origin, perhaps with an observer, but no galaxy. So the issue does not really exist.

"It starts out with D_proper negative. I thought you always start with the reference on the origin and the other one positive."

I've done this to let the main graphs of Figures 3 ad 4 progress in the same upward direction (it is after all the sane basic scenario, just over different timescales). The reference is still the origin...

-J

PS: S, do you now feel confident enough to take the "Cosmology-4-Engineers' exam?

Hi Jorrie,

"Only one button (galaxy), a tether and an origin, perhaps with an observer, but no galaxy. So the issue does not really exist."

Err, I see. I'll use the 'late a night' excuse again. So what did you use, a 'sky hook'?

"I've done this to let the main graphs of Figures 3 ad 4 progress in the same upward direction (it is after all the sane basic scenario, just over different timescales). The reference is still the origin..."

Right, origin is vertical, horizontal is ... <walks away mumbling>

"do you now feel confident enough to take the "Cosmology-4-Engineers' exam?"

But teacher, it can't be test time again yet! I may need more study time. Well, OK, bring it on, if I fail I fail.

-S

P.S. My dog ate my homework!

Hi S.

"P.S. My dog ate my homework!"

Ah, I think some extra homework should then be in order.

1. Two buttons, both presently at rest in the CMB frame (zero peculiar velocity) and 0.2 Gly apart, are suddenly tethered to each other with a 'perfectly rigid' tether. What will the initial force per unit of mass of the buttons be on the tether?

2. ...

OK, this is not easy, so do not try and give a numerical answer; just say what you think will happen.

-J

"1. Two buttons, both presently at rest in the CMB frame (zero peculiar velocity) and 0.2 Gly apart, are suddenly tethered to each other with a 'perfectly rigid' tether. What will the initial force per unit of mass of the buttons be on the tether? OK, this is not easy, so do not try and give a numerical answer; just say what you think will happen."

If I read your charts and descriptions properly, then in the case of the (0,0,0) universe there will be stretching force. In the (0.26,0,0.74) case there will be compression force up to 1.5 Gly, and then stretching force will take over. In the (0,0,0) case there may be a contradiction because in order have force between the buttons they would need mass, correct? But it is defined as having no mass.

-S

Hi S, OK, it was a trick question and you threw me back a 'trick answer'

No, the buttons are not massless, otherwise they must be treated as photons! We have to consider their masses negligible from a gravitational point of view, otherwise the dynamics differ from what we have here, but not massless.

Let's accept this for now and postulate what could have happened. Before tethering, the two buttons would be moving apart at H₀ D = 72/978 x 0.2 ~ 0.015c, i.e. very, very fast! It would hence have been impossible to tether them without something breaking. Before tethering, we would have had to attach rockets to the buttons (or at least to one of them) and boost them to temporarily have zero relative velocity.

Once tethered, we may read the strain gauges. You have it close, but not quite right. In the (0,0,0) case we would have read zero stresses in the tether, because the proper velocity of untethered buttons would have remained zero as in Fig. 2. In the (0.26,0,0.74) case, I have it that the compressing force would have lingered until around 7 Gy of age and then slowly reverse to a stretching force.

At the present time, that stretching force will be equivalent to 0.66 pico-g, or 6.6 x 10^-12 N per kg of button mass. Tiny! OK, replace the button with a galaxy and it is a different kettle of fish! The next blog will elaborate on this.

-J

Hi Jorrie,

I think you changed the scenario just to confuse me.

"It would hence have been impossible to tether them without something breaking. Before tethering, we would have had to attach rockets to the buttons (or at least to one of them) and boost them to temporarily have zero relative velocity."

We haven't worried about 'impossible' things before! If we assume an unbreakable tether, then as you implied there would be a lot of force! After the rockets (which countered the force), I agree there would be zero stress then. In the (0.26,0,0.74) case, I looked at d_proper but should have looked at v_proper which shows deceleration until around 7 Gy of age and then acceleration from then on. Do I get half credit?

-S

Hi S.

"Do I get half credit?"

Yes I think you deserve a lot of credit for persisting on these issues until resolved! You have this one perfectly correct now.

Now for the next question.

Still thinking about it...

-J

Hi Jorrie,

Wanted to return to the other post, and will, but I saw you posted something new and wanted to give it a read.

Let me start by saying that I really appreciate the equation section and I know it's probably a pain, but if you could include such sections more ofter I think it would be really helpful.

That said, can you, when you have a chance, name the cosmology variables/constants (H₀, Ω_m, a,etc.) Basically the variables that you know on sight but some of us may have to run off to wikipedia to remember what they are. For example (H₀ is the Time Independent Hubble Constant).

Ok, my thoughts on your post

I have a few questions.

You Wrote:

V_peculiar = L/√[L² + R²] and L = R₀ V₀/√[1-V_H₀²/c²] = constant

Shouldn't you use the time dependent hubble constant in the equation for conservation of angular momentum with respect to expansion above? I ask because in your "multiple epoch case" your peculiar velocity seems to decline nice and steadily despite the fact that expansion accelerates, then decelerates, than starts accelerating again. It seems like the decline in the peculiar velocity, while always declining, should decline faster, than slower, than faster again.

Also,

I think I understand V_peculair (velocity without expansion considerations or contributions) and I think understand V_hubble (velocity due only to expansion), but can you explain to me what V_proper is? I understand proper distance is the distance on a spacetime graph. I can imagine that V_proper is how distance changes over time on a spacetime graph. Fisrt, is everything I just said right? If it is, why does the proper velocity behave the way it does in your multiple epoch graph?

Hi Roger, you wrote:

"V_peculiar = L/√[L² + R²] and L = R₀ V₀/√[1-V_H₀²/c²] = constant

Shouldn't you use the time dependent Hubble constant in the equation for conservation of angular momentum with respect to expansion above?"

Since angular momentum L is a constant for a given particle, we just calculate it once, at the starting position of the experiment (R₀ & H₀ in this case). Thereafter L is used to obtain V_peculiar for other R, as shown.

You wrote: "It seems like the decline in the peculiar velocity, while always declining, should decline faster, than slower, than faster again."

It may be imperceptibly so, but in essence, the peculiar velocity is not directly influenced by the acceleration of expansion - it depends on 1/R alone, provided the velocities are non-relativistic, i.e., the de Broglie wave energy is small compared to the rest mass.

On proper velocity, yes, you have it correct. On the reasoning behind it, I thought I had it pretty well covered in the last three para's of the article. Remember that the button went from negative D, through the origin, to positive D. It behaves more or less like you wanted peculiar velocity to behave...

-J

PS: S is putting together a cosmological glossary, which should be a nice reference. However, I'll add a bit of definition to the OP.

You Wrote:"It may be imperceptibly so, but in essence, the peculiar velocity is not directly influenced by the acceleration of expansion - it depends on 1/R alone"

Yes but isn't 1/R directly related to the expansion rate, especially in the early universe? In your epochs the early inflationary period would have had tremendous 1/R changes. Isn't that true? I would think during that time we would see a sharper decline in peculiar velocity.

Hi Roger,

" Isn't that true? I would think during that time we would see a sharper decline in peculiar velocity."

Correct, but this is exactly what the graph shows. V_pec slope goes more vertical the closer you go towards time zero. What I implied is that it follows a precise 1/R curve, irrespective of expansion rate changes. This is required by the conservation of angular momentum principle.

Plotted against time, like I did, the curve is also virtually 1/t, with only tiny, imperceptible changes when the expansion rate changes.

-J

You Wrote:"Plotted against time, like I did, the curve is also virtually 1/t, with only tiny, imperceptible changes when the expansion rate changes."

Yes, I understand what you did. You plotted it as though there were no changes in the expansion rate. This isn't ok. The expansion rate of the early universe was too high for you to get away with that approximation. What you should have for your Peculiar Velocity graph is something that hugs the axis more due to the much stronger expansion rate in the early inflationary period (I realize some of this period didn't include matter, but it didn't stop on a dime when matter appeared).

1/R for peculiar velocity is an approximation that assumes no change in the expansion rate. I'm not arguing that the modern graph should be altered, for that one you are quite correct that the difference is trivial. However, for your "many epoch" graph I don't think it is and I don't think the approximation you're making is ok. Please think about it more.

Hi Roger, you wrote: "You plotted it as though there were no changes in the expansion rate. This isn't ok."

I think you are missing something here. Just to make sure, the first two charts are for the present and future epochs only (as I clearly indicated) and for simplicity do not include earlier epochs. The third chart is plotted from t = 0.2 Gy onward. All three use the full set of equations (no approximations) and the second and third have significant changes in expansion rate.

"1/R for peculiar velocity is an approximation that assumes no change in the expansion rate. "

Firstly, 1/R is a low speed approximation of the relativistic conservation of hyper-spherical angular momentum; it has little to do with whether or not the expansion rate changes.

Secondly, I have not used 1/R anywhere. It just so happens that relativistic conservation of angular momentum on the hypersphere does end up with something that resembles 1/R when the peculiar velocity is moderate.

I think now it's your turn to think some more.

-J

Jorrie,

Again. You Wrote:

L = R₀ V₀/√[1-V_H₀²/c²] = constant
V_peculiar = L/√[L² + R²]

In the second equation, how is R changing with time?

Roger

Roger, you asked: "In the second equation, how is R changing with time?"

By equations 7 and 8 of the OP equations list. Just integrate da = a H dt and you have a for any time you wish. Then R=aR₀, as per note (c).

H = H₀ √[Ω_m/a³ + Ω_v] sorts out the epoch dependencies, with decelerating expansion for the first 7 Gy and accelerating expansion thereafter.

-J

Jorrie,

Can you show the plot of R over time (for the .2 GY starting point case)? I think that would be helpful.

Roger

Hi Roger,

Plenty of this sort in Relativity 4 Engineers, but here is a more colorful one.

This one is plotted for H₀ = 71.9 km/s/Mpc, 26% matter and 74% vacuum energy components.

The present age is 13.66 Gy, with R = 100 Gly, which is an arbitrary scale for convenience. It is for a flat universe, where the actual radius should be infinite.

As you can see, it is not a very remarkable curve at all! Right at t=0.2 Gy, the slope is quite large (some 8:1), but only for a very short time.

-J

You Wrote:"As you can see, it is not a very remarkable curve at all! Right at t=0.2 Gy, the slope is quite large (some 8:1), but only for a very short time."

So if you used this varying radius over time (R) to construct your peculiar velocity curve it should be correct then, which is what I think you are saying you did.

The curve doesn't appear to be remarkable because of the long timescale. For most of the history of the universe, d²R/dt² has been small. If you were to restrict the curve from .2 Gy to .5 Gy, I imagine it would be quite remarkable since the d²R/dt² over that time was large.

I'm intrigued by the massive decline in momentum that occurred in the early universe (< 1 My). Every massive particles must have been traveling at extremely high speeds (like .999 c or something).

I look forward to your post on the Cosmic Tidal Force.

Hi Jorrie,

Hi Jorrie,

I've been away from your blog for a while.

I may be reading your Multiple epoch case graph wrong. You say:

"At cosmic time t = 0.2 Gy after the BB, the tethered galaxy was at D = –0.1 Gly.^[1]".

Since the units of distance in the graph are Gly/10, it looks to me like the galaxy was at D = -.001 Gly at around t = 0.2 Gy.

. . . . . . . . . .

I wonder if it is possible to construct any version of this (0.26, 0, 0,74) case where the tether is cut at t = 0.2 Gy and the galaxy never passes through the origin. That is, the galaxy approaches the origin at first under the influence of gravity, but before it reaches the origin it stops and begins moving away from the origin under the influence of vacuum energy. Obviously that suggests untethering the galaxy at some much larger distance from the origin.

Although V_peculiar diminishes at the rate of 1/a and the expansion rate always remains positive, this does not necessarily preclude V_proper from changing sign. If at some point V_Hubble (that is, HD) begins to exceed the contemporaneous peculiar velocity (V_peculiar), then the sign of V-proper will change. But as long as D is decreasing, it partially or fully offsets the contribution of the vacuum energy toward increasing H, so it may take a very long time for the sign of V_Hubble to change, if it changes at all. So I'm not sure whether there can ever be enough time in the .02 Gy untethering scenario for the sign to change before the galaxy reaches the origin, even if the initial untethering distance approaches infinity.

Of course if the untethering occurs much later, it seems probable that V_proper can change sign before the galaxy reaches the origin. For example, if the untethering occurs just before the acceleration of the expansion becomes zero, and the untethering distance is sufficiently large.

. . . . . . . . . .

On a related subject, I think it would be instructive if you were to make a graph showing the wordlines for a "double-tethered galaxy" exercise. This is an exercise where a first button (galaxy) is tethered to the origin, and a second button (galaxy) is tethered to the first button. Both tethers are released at the same proper cosmological time.

In the matter-only case, I expect the paths of the two galaxies should always cross exactly at the coordinate origin, regardless of the ratio of the two galaxies' initial distance from each other relative to the first galaxy's initial distance from the origin. (The galaxy that initially was more distant will cross the origin at a higher V_proper than will the other galaxy). After crossing the origin, each of the galaxies will asymptotically rejoin its local Hubble flow at points at the same proper distance from each other (but opposite sign) as they were at the time of untethering.

I'm not sure whether a similar symmetry applies in a perpetually-expanding model that includes lambda, (i.e., your multi-epoch case). If the paths of the two galaxies eventually cross, do they cross at the origin? (I think the answer is yes.) Does the distance between the galaxies always retain a fixed ratio relative to the contemporaneous distance between one galaxy and the origin? (I think yes). If so, that ratio should hold even if the V_proper of both galaxies changes sign and the paths of the two galaxies never cross each other or the origin.

Jon

Hi Jon, good to hear from you again.

No, I meant that the actual value is divided by 10 before plotting (i.e., I plotted D/10), so D actually starts at -0.1 Gly. I had to do it to fit the distance onto the scale of v/c.

When I have time, I'll dig up the old spreadsheets again and see if I can plot the scenarios that you proposed.

-J

Hi Jorrie,

OK, I understand the scale on your graph now.

By the way, regarding the possibility of V_proper changing sign, the maximum length of the tether must be constrained such that V_peculiar remains <c. In other words, the tether length cannot exceed the Hubble Radius. Other than the V_peculiar constraint, there is no constraint on the magnitude of V_proper itself, which may be >c.

Jon

Hi Jon,

Here is a plot of the relativistic situation for a button untethered at D ~ -0.3 Gly, at t = 0.2 Gy, Vpec ~ 0.999c

Peculiar velocity decays as per Tamara Davis's Fig. 3.5, (not 1/a), taking into account that she plotted for a different mix of energies.

The button crosses the origin at age t ~ 0.66 Gy, against the t ~ 1.4 Gy crossing of the D = -0.1 Gly case presented in the OP. I'm still not sure why this difference arises, but it probably has to do with simultaneity issues, which are very tricky in a relativistic cosmos.

-J

Hi Jon.

I played with the spreadsheet again, but I think you have answered most of the questions already!

"I wonder if it is possible to construct any version of this (0.26, 0, 0,74) case where the tether is cut at t = 0.2 Gy and the galaxy never passes through the origin."

I could find no distance at which this would happen. At 0.2Gy, a~0.05, H~3.3 Gy^-1, meaning the Hubble radius was only ~0.3 Gly, which would have been the maximum tether length before peculiar velocity exceeds c. As you also said, it can obviously work for un-tethering at a much later epoch, just before vacuum energy dominance (~7 Gy).

"I think it would be instructive if you were to make a graph showing the wordlines for a "double-tethered galaxy" exercise. This is an exercise where a first button (galaxy) is tethered to the origin, and a second button (galaxy) is tethered to the first button. Both tethers are released at the same proper cosmological time."

One would surely expect that the two galaxies would cross exactly at the origin, if the tether lengths are short enough so that they cross at all (in a multi-epoch case). I did not simulate this directly, but indirectly, by changing the tether lengths on a single galaxy and checking at what time the origin is reached. To my surprise the time depended on the tether length - the longer the tether, the earlier it crossed the origin (at least that was according to my model).

This does not sound quite right, because if we take galaxy A at 0.1 Gly and galaxy B at 0.2 Gly, we could have translated the origin to galaxy A and viewed the original origin as galaxy C. B and C are now equidistant from A, in opposite directions and we could expect them to 'fall' towards the origin identically. Closer inspection revealed that it is the relativistic equations (3) and (4) (relativistic angular momentum) in the OP that cause the asymmetry. If I replace them with straight Newtonian versions, the galaxies cross the origin at the same time...

I'm still puzzling as to whether there is perhaps a change of inertial frame issue between the two origins, causing the asymmetry and hence that the intuitive approach could be wrong. Will come back on that...

-J

Hi Jorrie,

Rargarding V_proper changing sign, you agree that it can't happen if the untethering is at 0.2Gy, but it can happen if the untethering occurs at some later time before the matter-dominated epoch ends. I further predict that there is some "intermediate" untethering time such that the galaxy approaches all the way until it is infinitesimally close to the coordinate origin, then reverses course and accelerates along that opposite course forever.

Intuitively, V_proper cannot change sign if the galaxy actually reaches the origin, because there is exactly zero acceleration acting on the galaxy at the origin, so it must continue in its previous direction, and as it moves away from the origin, it will experience additional acceleration in the direction it is already traveling. [Edit: Maybe this isn't true, at least in theory. If the universe is still matter-dominated when the galaxy passes the origin, it will experience gravitational acceleration toward the origin which exceeds the vacuum energy acceleration away from the origin. Maybe a scenario could be finally tuned such that the galaxy passes the origin, changes direction again, and continues to occilate back and forth across the origin until the matter-dominated epoch completely ends; then it should begin accelerating continuously away from the origin.]

You said:

"To my surprise the time depended on the tether length - the longer the tether, the earlier it crossed the origin (at least that was according to my model). ... Closer inspection revealed that it is the relativistic equations (3) and (4) (relativistic angular momentum) in the OP that cause the asymmetry. If I replace them with straight Newtonian versions, the galaxies cross the origin at the same time..."

I agree that your spreadsheet is calculating an incorrect result, and I believe that as you suspect the problem lies with your equations (3) and (4). The problem is that SR cannot be applied to calculate a non-local peculiar velocity in FRW coordinates. SR does apply to peculiar velocities, but only exactly locally at the spacetime location where the moving object is passing a fundamental observer (that is, the fundamental observer itself has zero peculiar velocity relative to its local Hubble flow.)

And of course the distance to a passing object at zero distance from a fundamental observer will experience zero Lorentz contraction. So SR falls out of the calculations. In a flat FRW model at critical density, peculiar velocities are calculated in a linear manner, with no Lorentz transformation. Happily there is no spatial curvature in that case to mess up the simple linearity of distances.

Also, there is no time dilation as between two fundamental observers (e.g., one at rest at the origin, and another at rest in its local Hubble flow at the contemporaneous location of the moving object) because all FRW fundamental observers share a common cosmological proper time (i.e., time from the Big Bang as measured on their respective wristwatches) at least in the case of a model at critical density. There is of course the normal SR time dilation when one shifts between the respective inertial frames of the local fundamental observer and the passing object at that same location, but that doesn't factor into the calculations we are working on.

Jon

Hi Jon.

I agree with your first part, but I'm not yet convinced that the relativistic treatment of peculiar velocity (in the way Davis and myself have done) is in error.

Right is the Newtonian treatment of the exact same scenario as in #33 above, starting at Vpec ~c. Note how Vpec decays with the standard, low velocity 1/a profile, which I believe is incorrect for a super-relativistic particle.

You wrote: "The problem is that SR cannot be applied to calculate a non-local peculiar velocity in FRW coordinates. SR does apply to peculiar velocities, but only exactly locally at the spacetime location where the moving object is passing a fundamental observer (that is, the fundamental observer itself has zero peculiar velocity relative to its local Hubble flow.)"

But, Vpec as plotted is local for every co-located fundamental observer that the button passes. It is calculated from particle momentum decay (which is the same as angular momentum conservation in a hyper-spherical model). Tamara Davis uses the same equation as my (3) and (4), just expressed slightly differently (in terms of a, not R).

You wrote: "In a flat FRW model at critical density, peculiar velocities are calculated in a linear manner, with no Lorentz transformation. Happily there is no spatial curvature in that case to mess up the simple linearity of distances."

OK, this may be a source of error in my model, because I added V_pec and V_Hubble relativistically to get V_prop. It however does not influence V_pec, which should still fall out of relativistic momentum decay. I changed the V_prop calculation to linear and indeed, it now has a more "physical look and feel", as shown below. It originally shot up to V_prop ~ c almost instantly, where it now takes the more usual acceleration path. In fact, V_prop now reaches c ~ 1 when the button passes the origin, which could perhaps be expected, since it starts 'almost like a photon' (V_pec=0.999c).

However, the crossing point is still at ~0.67 Gly, so buttons tethered at -0.1 Gly and -0.2 Gly will still not cross at the origin. It means the relativity of simultaneity still plays a role here - after all, the two buttons are fast moving through their local space (relative to fundamental observers).

My current (relativistic) thinking is as follows. Let us call the origin observer O, with A and B the button observers at -0.1 and -0.2 Gly respectively. All fundamental observers will agree that the tethers are cut simultaneously (or rather that the tethered buttons start to 'move' simultaneously), but A will not agree that O and B started to 'move towards her' simultaneously. She will observe B to start moving before her tether is cut.

Hence, there is no paradox if O and B, being equidistant to A, do not pass A simultaneously. We must not forget that B and C are not inertial observers, due to the tethers, which make them experience severe tidal forces.

Does this make any sense, Jon?

Hi again.

Here is a rough 'simultaneity sketch' of what I meant by: "All fundamental observers will agree that the tethers are cut simultaneously (or rather that the tethered buttons start to 'move' simultaneously), but A will not agree that O and B started to 'move towards her' simultaneously. She will observe B to start moving before her [O-side] tether is cut."

The black circle segment is the line of simultaneity for fundamental observers (cosmological time). The up-arrows are button world-lines while still tethered.

The red circle segment is a rough line of simultaneity as per A immediately after being untethered (inertial now, but proper speed still zero). It is clear that the B event happens before the O event, from this A POV.

I used the hypersphere simply for ease of visualizing Hubble flow relative to peculiar flow. My charts are actually for flat space. The ordinality is the same if drawn in on a flat Minkowski, but the constant (tethered) proper distance is not so visible.

Hmmm, yes Jorrie this is a more complex problem than it first appeared.

The acceleration of V_proper in your "improved" relativistic chart looks much more natural than the original version. I think this version is correct.

I think the difference between the SR and Newtonian versions arises entirely from the different rates of decay in V_peculiar, and the lack of symmetry between the B-A tether and the A-O tether. As you emphasized, the longer the tether, the more relativistic the speed, and the slower the rate of decay in V_peculiar, so the sooner the galaxy reaches the origin. There is no Lorentz contraction involved.

I am skeptical about the 'relativity of simultaneity' explanation you graphed. A, B, and O are all fundamental observers, at rest in their own local Hubble flow(s). They share the same cosmological proper time, so they should all observe collocated-events to happen at the same time (after correcting for light travel time).

Assume that by prearrangement, A, B, and O each fire lasers to cut the passing tether(s) at their own location at the same cosmological proper time. Then A will observe the distant laser flashes from B and O at the same time (assuming A is equidistant from B and O), after the appropriate amount of light travel time (so of course A observes the B & O flashes after observing his own flashes).

Observers A, B and C will observe that the effect of the untethering of the A, B and O galaxies begins simultaneously, but thereafter the three galaxies will behave non-symmetrically. At the instant of untethering, galaxy B already has a highly relativisitic V_peculiar of .999 c toward observer A, so galaxy B's V_proper will begin accelerating toward A (as the cosmic expansion rate slows and its V_Hubble relative to the origin decreases).

Meanwhile, at the time of untethering, galaxy O has a V_peculiar of zero, so its motion will be governed entirely by the Hubble flow. Galaxy O will immediately begin moving away from A at its relative Hubble velocity (i.e., initially its V_proper is nonzero). Galaxy O was already moving away from observer A before the untethering, so galaxy O does not actually accelerate upon untethering. Galaxy O is the only galaxy that remains permanently colocated with its inital observer.

Meanwhile, at the time of untethering, galaxy A has a semi-relativistic V_peculiar of .449 c away from observer A and toward observer O. Galaxy A needs to travel only half the distance as galaxy B to reach the origin, and galaxy A begins with twice the V_peculiar. However, because galaxy A's more relativistic V_peculiar decays at a a much slower rate than galaxy B's less relativistic V_peculiar, all 3 observers will agree (after correcting for light travel time) that galaxy B will overtake galaxy A before it arrives at the origin. Galaxy B arrives at the origin well before galaxy A does.

The effect of swapping the coordinate origin is asymmetrical because the degree of "relativisticness" of the 3 galaxies' respective peculiar velocities cannot be made symmetrical relative to different fundamental observers.

Galaxy B will overtake galaxy A exactly at the origin only if their initial V-peculiar's are entirely non-relativistic, so at best it happens asymptotically close to the origin rather than exactly at it. Relativistic massive particles are messy!

Jon

Hi Jon, you wrote: "I think the difference between the SR and Newtonian versions arises entirely from the different rates of decay in V_peculiar, and the lack of symmetry between the B-A tether and the A-O tether. ... There is no Lorentz contraction involved."

Remember that my Eqs. (3) and (4) sport relativistic time dilation (gamma) and hence Lorentz contraction is inherently there as well (and so is the relativity of simultaneity). In SR, you cannot have the one without the others. It does not mean A observes O and B at different distances - they are equal, just different to what a fundamental observer would have observed.

Your: "I am skeptical about the 'relativity of simultaneity' explanation you graphed. A, B, and O are all fundamental observers, at rest in their own local Hubble flow(s)" is not what I meant by the "button observers". In my original sketch, only O is a fundamental observer. The dotted arrows (right, A_f and B_f) indicate the co-located fundamental observers. Sure, they will agree on the time that the tethers are cut, but not so for A and B, who are relativistically moving relative to A_f and B_f.

Since the relativistic decay of particle momentum inherently includes time dilation, which implies the relativity of simultaneity, I suppose the two explanations (yours and mine) are equivalent anyway. Your way is perhaps easier to use and understand.

"Galaxy B will overtake galaxy A exactly at the origin only if their initial V-peculiars are entirely non-relativistic, so at best it happens asymptotically close to the origin rather than exactly at it. Relativistic massive particles are messy!"

Yup!

In any case, thanks for participating again, Jon - it helps a lot if someone with your insight comments on technical work done...

-J

PS: the charts of the OP do not change in any way, because they are not very relativistic. I've just changed Eq. 10 to the Newtonian form.

Hi Jorrie, you wrote: "Remember that my Eqs. (3) and (4) sport relativistic time dilation (gamma) and hence Lorentz contraction is inherently there as well (and so is the relativity of simultaneity)."

I was unclear what the L in your Eq 3 represents, but from the reference to Tamara Davis I see it represents relativistic momentum. This does indeed reflect a time dilation in the moving galaxy's rest frame, relative to cosmological time. But as I said earlier, Vpec is always measured "officially" only at zero distance, meaning that the Lorentz contraction is zero. You are correct that in theory there is Lorentz contraction, but the amount is always zero.

Since at relativistic speeds, the relativistic momentum exceeds the Newtonian momentum in a non-linear way, I can see why the Vpec of relativistic massive particles decays at less than the Newtonian 1/a rate.

I agree that there is relativistic time dilation and failure of simultaneity in the rest frames of galaxies A & B relative to the cosmological proper time of fundamental observers. But I also think it is appropriate to examine the whole scenario in terms of cosmological proper time only. In which case, as discussed, relativistic galaxies A & B do not cross the origin at the same time. From my perspective there is no contradiction (e.g. a violation of symmetry) that needs to be explained by reference to an SR failure of simultaneity. But I agree there is a failure of simultaneity in the relativistic galaxy A & B rest frames.

Enough with massive particles! I am more interested in photons. Next I'd like to offer some thoughts about cosmological redshift.

Hi Jon.

I would appreciate your inputs on cosmological redshift. A nice place to add your comments would be in my previous Blog post on Redshift. I pose the following questions there:

"In the time that the CMB photons were in flight, the balloon expanded by a factor 1089 and the photon wavelengths were 'stretched' by a factor 1089, giving their redshift as: z = λ/λ₀ - 1 = 1088. Quite reasonable, it seems at first sight,^[1] but how can a photon's wavelength be stretched? A single photon does not even have a defined size, so stretching it is not conceptually very palatable!

"Another reasonable explanation may be that it is just different frames of reference between transmission and reception of the photon and that the redshift is a coordinate transformation issue, resulting in Doppler shift. However, the two frames of reference may be moving away from each other at greater than the speed of light in vacuum (c), yet we still measure a real, finite redshift. How do we reconcile this fact with the Doppler shift equations of Einstein, which do not work for recession speeds equal to or larger than c?"

I then offer a "balloon analogy" explanation...

-J