I don't think it's a meaningless debate at all, and you just proved that. Whether or not both models are correct, cancel each other, or lead to further investigation, it adds fuel to the concept of, and visualization of 'Infinity' having an 'n' dimensional nature. Your legwork in calculating could shift the paradigm enough to require further investigation. Because I lack the mathematical skills (as do others), I shall ponder this as a philosophical question, for the entertainment and enlightenment of the right brained rabble. Thank you.

Carl

Hi Jorrie,

Great chart! You are the master of turning these confusing ideas into instructive pictures. And no, I haven't tried fooling around with Lambda because this subject is confusing enough at this exploratory stage without introducing another acceleration vector.

One conclusion to be drawn from the analysis in the chart is that in the simple case of a matter-only model, the Robertson-Walker metric can be interpreted as nothing more than the Schwarzschild metric with changing density added in. (Except of course for the fundamental difference in how superluminal motion and density above 2GM/r are treated). The change in density being proportional to the change in volume traced by any given change in radius.

The SR time dilation in the kinematic model makes you uncomfortable. You can avoid this problem if you accept my proposition that the gravitational time dilation modeled by the shell theorem exactly offsets the SR time dilation in the flat space scenario. If "phantom" SR time dilation is acceptable in cosmologically-sized spheres of matter, then why not "phantom" gravitational time dilation? As I said it's compatible with FLRW because the two effects offset each other.

I've been imagining a new chart that would shed light on the balance between SR and gravitational time dilation ("TD"). I think this chart would help illustrate the correspondence between the FLRW and Schwarzschild metrics, because both are incorporated in it.

The x-axis shows density at a single point in time, in units of 2GM/r, from 0 to 1 and even beyond 1. The y-axis shows recession velocity at a single point in time, in units of c, from 0 to 1 and beyond 1. A 45 degree straight line from the origin represents escape velocity. Various hyperbolic lines starting at the origin and curving back toward the x- or y-axis show lines of equal TD (and spatial curvature?) for various combinations of SR and gravitational TD that do not equal escape velocity. The more such a line curves away from the ecape velocity line, the more that space and time are curved. The upward curving lines signify an "open" range where the SR TD exceeds the gravitational TD and the spatial curvature is negative (hyperbolic). The downward curving lines signify a "closed" range where the gravitational TD exceeds the SR TD and the spatial curvature is positive (hyperspherical). The (0,0) origin represents an SR inertial frame, where space is flat (because the escape velocity line intersects the origin) and the time dilation is 0. The y-axis itself represents an empty universe (e.g. a Milne-like model) at various recession speeds, with the maximum possible hyperbolic spatial curvature. The x-axis represents a current or future black hole or cosmic singularity, with the maximum possible hyperspherical spatial curvature. (At the origin the maximum possible curvature is zero.)

One interesting observation from this chart I imagine is that, assuming a universe which is Ω=1 and spatially flat, one can select any arbitrary constant radius of cosmological size, and plot out points on the history of our universe. The Big Bang (or end of inflation) would be infinitely to the right and up along the escape velocity line, and the plot points would slide down along the escape velocity line toward the origin as time progresses. This illustrates how the FLRW model eventually slides effortlessly from superluminal motion and > BH density into the "normal" Schwarzschild-permissible range.

Jon

Hi Jon.

I will reply fully later, but in the meantime I've found the equation for the 3D kinematic acceleration in proper distance coordinates for the ΛCDM case in Peebles:^[1]

d²r/dt² = -4/3 Π r G ρ₀ / a³ + 8/3 Π r G ρ_Λ

where r is the proper distance from the origin, ρ₀ is the present total density (including vacuum energy) and ρ_Λ is the constant vacuum energy density.

One can make an interesting observation from the equation. The negative term shows that deceleration component caused by of the "mass of the vacuum" works just like ordinary matter, being diluted by the increasing volume (the 1/a³). As expected, the accelerating effect of the vacuum energy does not get diluted by volume and increases without limit with r.

The chart shows the usual culprits for the (0.3,0.7) case and once again the hyperspheric and kinematic acceleration curves sit on top of each other.

Another potential incompatibility cleared. I have not checked what happens for other types or dark energy, i.e., with equations of state w <> -1, but I suspect they will also hold. I will look into the time dilation issue next.

-J

[1] Slightly reworked from Peebles (1993) eq. 13.3, p. 312.

Hi again Jon, on the time dilation problem you wrote:

"You can avoid this problem if you accept my proposition that the gravitational time dilation modeled by the shell theorem exactly offsets the SR time dilation in the flat space scenario."

I still can't buy the "offsets each other" proposition, because while they are equal, they are additive, not subtractive - as we have agreed before. I want the clock at the origin to tick at the same rate as the clock "moving" with the Hubble flow at shell radius r, like it is in the hypersphere model. It does not help that a clock sitting stationary on the shell at r is ticking at the same rate as a clock that moves at the escape velocity for r, but sitting outside the cloud at infinity (the flat spacetime that you mentioned, if I understood correctly).

In your frictionless dust cloud, Schwarzschild says that the clock at the origin ticks at

dΤ₀² = (1 - 3GM/r) dt², where dt is a stationary distant clock's time differential and M is the total mass inside the sphere with radius r.

Likewise, the clock moving at r at escape velocity v_e² = 2GM/r ticks at

dΤ_R² = (1 - 2GM/r - v_e²) dt² = (1 - 4GM/r) dt², if we take the weak field approximation. This is clearly not equal to the rate of the clock at the origin. The difference gets even larger when a strong field regime is considered.

How do you reconcile this discrepancy in a rigorous way?

I will come back to your new chart idea if I can reconcile myself with this discrepancy.

-J

Hi Jorrie,

Sure, I understand that the problem boils down to whether to apply the internal or external Schwarzschild solution to calculate the cosmic phantom time dilation. And I've said that on the surface it seems like the internal solution fits the situation best.

But after stewing over this a lot, I've changed my mind, mostly. Recall that the primary purpose for which we're employing the metric here is as a component that contributes to the cosmological redshift we observe (by offsetting the SR time dilation). In the kinematic model, we exclude any effect from 'expanding space', so for the current discussion we are comparing conditions as they existed at the emitter at the time of emission to conditions now at the observer's location. Thus the time dilation involves a density comparison as between two widely separated points in time.

My calculation of the internal Scwarzschild metric for our observable universe (i.e., within our particle horizon) now, with us being the center (origin), is that the gravitational time dilation factor here, for an infinitely distant observer at a zero-density location, is very small: about 1.0160. In other words, the gravitational time dilation of the dense past compared to the very undense present is quite close to comparing the dense past to zero-density infinity. So I think we can temporarily set that tiny discrepency aside, and revisit it later as a potential slight correction. Which says that it really is very close to correct to apply the Schwarzschild external metric to calculate the phantom gravitational component of redshift.

Despite our ability to use the external solution, we must always keep in mind that the internal solution is really the correct one. This makes my brain hurt trying to think about the implications:

1. Since that 1.0160 discrepency means that the gravitational redshift doesn't exactly cancel out the SR time dilation, it suggest that there may be a proportional error in the 1:1 redshift-expansion correlation. Distant galaxies are slightly less distant than their observed redshift indicates. Also, there is a corresponding slight negative (hyperbolic) spatial curvature, which suggests that cosmological features (such as the quadrapole moment of the CMB fluctuations) are slightly larger than observations indicate.

2. Because the gravitational time dilation was even less of a contemporaneous factor in ancient eras, it also suggests the possibility that the universe was characterized by significant contemporaneous negative (hyperbolic) spatial curvature and SR time dilation in ancient eras (i.e., within an ancient era rather than across different eras.) Even though the universe is (or may be) spatially flat today. This is contrary to the normal principle that spatial curvature increases as a function of time. This ephemeral past spatial curvature might have left its own imprint on the redshift and feature sizes we observe now, but I'm not sure, maybe not.

3. This even suggests the possibility that the FLRW metric is an incorrect model for the evolution of the universe. Instead, we should develop a version of the RW line element derived from (or partially covariant with) the internal Schwarzschild line equation rather than from the external one. Of course I don't know at this stage whether such a solution would be an exact solution for the Einstein Field Equations, but hey, why not?

I have no idea whether any of these ideas would result in calculations that differ significantly from observations and current theoretical predictions, in which case they may just be wrong. But one must be careful to ensure that all of the theoretical effects of a different model are well understood before trying to compare calculations to observations.

By the way, in suggesting that the external Schwarzschild metric be applied as an approximation, I see no logical problem with the fact that it is measuring time dilation relative to hypothetical observers at zero-density infinity, that is, hypothetical observers located outside the FLRW universe. I don't see anything in the FLRW metric that prevents one from modeling an open, expanding, finite FLRW "dustball" residing in otherwise empty space. It's analogous to the fact that we create hypothetical observers at zero-gravity infinity for the Schwarzschild metric, which isn't actually possible in the real world either.

A couple more comments about the time dilation chart I envisioned:

- The x-axis does not necessarily represent a present or future black hole or singularity. It just represents a present or future collapse of the entire dust mass into a single central mass. Which may or may not have the right characteristics to further collapse to a BH.

- There is a large "exclusion zone" in the chart, out beyond the "1" coordinate on the x- and y-axis. It is impossible for the universe to exist in the regions "outside" of a line (probably an inward hyperbolic curve) drawn from the 1 coordinate on each axis and assymptotically approaching the escape velocity line at infinity. This illustrates the accepted concept (mentioned above) that the very early universe needed to be extremely flat (i.e., close to escape velocity) in order for the present universe to be anywhere near flat. The lower exclusion region represents the expanding universe quickly recollapsing into a BH, or failing to launch. The upper exclusion zone represents a universe expanding so rapidly that there may have been no BB at all.

Welcome to my nightmare!

Jon

Hi again Jon.

Before replying to your 'nightmare', a clarification is needed. You talk about the "internal or external Schwarzschild solution". If by 'internal' you mean inside the event horizon, then your origin is at the central singularity, where all bets are off, because Schwarzschild breaks down. Hence, you cannot compare time there with time anywhere else.

I still do not quite understand which clock your are comparing to which...

-J

Hi Jorrie,

No I don't mean inside the event horizon.

I mean the Schwarzschild solution for gravitational time dilation at the center (not the surface) of a diffuse body (like a star) as measured by an observer at zero-density infinity. The relativistic formula being:

3/2 * (1-2GM/c²r)^1/2 - 1/2

Jon

Hi Jon, OK the equation you gave is a good enough approximation for the weak field of Friedmann. The rigorous one is as I gave in #8, with the c² inserted (which I neglected):

dΤ₀² = (1 - 3GM/(c²r)) dt², where dΤ₀ is the time differential at the origin and dt is the time differential for an observer at your zero-density infinity (I have no problem with that concept).

To proceed, can you give me a numerical example of how you see the cosmological time for (say) an observer just within our particle horizon, compared to our time? Maybe I will comprehend better where you are going with this.

-J

Hi Jorrie,

First, it may help if I use the correct terminology, they are called the "interior" and "exterior" Schwarzschild solutions, not "internal" and "external."

What is the source of your equation? I don't recognize it. The equation I gave is the exact relativistic solution, not merely a weak field approximation. The complete relativistic interior solution is:

dT/dt = 3/2 * (1-2GM/c²R)^1/2 - 1/2 * (1-2GMr²/c²R³)^1/2

See, e.g., Gron & Hervik eq 10.265

At the center, r = 0, so the equation reduces to the simpler version in my last post. At the "surface" (outer radius), r = R, and the equation reduces to the normal exterior metric. Contemporaneous time dilation at the surface relative to the center (I think that's the question you asked) becomes:

dT_surface / dT_center = (1-2GM/c²R)^1/2 / (3/2 * (1-2GM/c²R)^1/2 - 1/2

By the way, the absolute equivalence of an instantaneous snapshot of a homogeneous, flat, matter-only FLRW model with the Schwarzschild exterior solution is required by Birkhoff's theorem. Which says that the gravitational field exterior to any spherically symmetrical mass configuration (whether or not in motion) is both stationary and governed by the exterior Schwarzschild metric.

. . . . . . . . . .

Well, I went over my math again, and discovered I had an error in my spreadsheet (as you may have suspected.) Now it turns out I can't actually calculate the time dilation for our observable universe now or in the past, because 2GM/c²R > 1 and the equation blows up. (The present ratio ~ 11.) Drat!

So I am limited to running the calculations only inside the normal Schwarzschild range, c < 1 and 2GM/c²R < 1. Well, at least the "exterior" gravitational and SR time dilation equations are exactly equal and opposite at escape velocity both inside and outside the normal range. But my rationale for using the exterior metric instead of the interior one has vaporized. Back to the drawing board (This really is a nightmare!)

So here's different rationalization for applying the exterior metric. The exterior metric combined with SR at escape velocity tells us that, at the emitter's emission distance r from us, space is locally flat and there is no net time dilation, as measured by a hypothetical observer at zero-density infinity. The redshift of the light that such an external observer would (if he could) receive from the emitter would therefore be unaffected by time dilation. By the same token, the external observer could consider the emitter to be at the "center" and us to be out at the radius. He then would observe light emitted from our galaxy to also be unaffected by time dilation. The external observer therefore can reasonably conclude that, if light from neither we nor the (original) emitter is time dilated when he receives it, then we and the original emitter are not locally time dilated relative to each other. He can apply this conclusion to any pair of emitters in the universe at any distance apart. If we and the emitter are not locally time dilated relative to each other, then no artifact of either gravitational or SR time dilation should appear in the light we receive from each other.

This logic sounds a bit bootstrapped, but that doesn't mean its wrong. It's straightforward Schwarzschild logic applied without any reliance on FLRW. In order to keep this "pure" Schwarzschild, the observer's galaxy mass technically needs to be outside the radius used in the calculation, so that it can remain a single-mass vacuum solution. The observer's galaxy mass can be infinitesimally close to the radius, as long as it's outside, and the observer is located exactly at the radius.

Jon

Hi Jon, .

Yes, the terminology can be very confusing. Some authors use the term "interior" and "exterior" simply in relation to black holes, where "interior" then meaning inside the event horizon. They sometimes add qualifier "black hole", e.g. "Interior Schwarzschild black hole solution", in http://arxiv.org/abs/gr-qc/0609042 (Rosa Duran et. al). Your equation is correct for the interior of a non-black hole. I confused myself with a mix of black hole interior and star interior.

I'm still not sure why you want to go this very difficult route. In the "Root of all Evil" Francis et. al wrote: "The metric of spacetime in the region of a galaxy (if it could be calculated) would look much more Schwarzchildian than FRW-like, though the true metric would be some kind of chimera of both." Sounds somewhat like your "nightmare". The standard FLRW view with the hypersphere that could have an infinite (undefined) hyper-radius for the "flat" case, or even an complex number for hyper-radius in the "open" case, is rather straightforward and works everywhere for all cases, Lambda included!

I still do not follow what you wrote here: "The exterior metric combined with SR at escape velocity tells us that, at the emitter's emission distance r from us, space is locally flat and there is no net time dilation, as measured by a hypothetical observer at zero-density infinity" (and what follows on that).

Please show a clear calculation and I'll maybe understand what you mean. If I reply while confused about your drift, the confusion will just increase.

-J

Hi Jorrie,

Here's another interesting scenario to apply both the 'expanding space' and 'kinematic' models.

Imagine there are two galaxies in the tethered galaxy problem. The first galaxy is tethered to the origin. The second galaxy is tethered to the first galaxy. Both tethers are the same length.

First let's look at the 'expanding space' explanation. At the instant of untethering, the proper velocity is zero as between the two galaxies themselves and relative to the origin. Galaxy #2's peculiar velocity is twice Galaxy #1's, and relative to the origin the local Hubble flow at Galaxy #2 is twice that at Galaxy #1. I think that, as the Hubble rate decays with time, the proper distance between the two galaxies will decrease, because Galaxy #2's peculiar velocity is twice Galaxy #1's. I think they will exactly converge at the origin. Thereafter, Galaxy #2 immediately will pass by Galaxy #1, and the distance between them will progressively increase again, until they both asymptotically come to rest in the Hubble flow, at respective proper distances equal to (but at the coordinate negative of) their original distance from each other at the instant of untethering.

In the kinematic model, on the inbound leg Galaxy #2 is always at a greater radius from the origin than Galaxy #1, so the sphere of matter defined by its radius, and therefore the gravity it "feels" toward the origin, is always larger than that of Galaxy #1. So Galaxy #2 experiences more gravitational acceleration on the inbound leg, and more (but not very much, due to the declining cosmic density) gravitational deceleration on the outbound leg.

So the result is directionally the same in either model.

. . . . . . . . . .

One can generalize this scenario to conclude that a spherical ball of homogeneously dispersed dust particles which start at proper rest relative to each other will collapse to its center point, and if the dust particles somehow avoid colliding, the dust ball will proceed to turn inside out and its proper radius will continue expanding thereafter for at least as long as the expansion of the universe continues.

. . . . . . . . . .

Now let's change the scenario so that at the start, instead of the galaxies approaching the origin, the are moving away from it. First test Particle #1 is fired out of a cannon at the origin, at proper velocity v. Then after elapsed time t, test Particle #2 is fired out of the same cannon at the same proper velocity. So at the instant Particle #2 is fired, its proper velocity relative to Particle #1 is zero. (Actually it's less than zero but we'll assume that the negative relative velocity at that instant is insignificant).

In the 'expanding space' model, Particle #2 again has a higher peculiar velocity in its local Hubble flow than Particle #1 has in its own. So that as the Hubble rate decays with time, Particle #2 again will gain on Particle #1. I'm not sure whether Particle #2 will eventually catch up with #1, or will approach it only asymptotically. Probably the latter. This analysis again seems to be duplicated by the kinematic model, with Particle #1 being subject to a larger gravitational sphere than is Particle #2.

Francis, Barnes et al evidently bollix up the 'expanding space' interpretation of this "outbound" scenario at the end of section 2.6.2 of their paper "Expanding Space: the Root of all Evil?" They claim that the two test particles "being subject only to expanding space, [will be stretched apart] in proportion with the scale factor. These are essentially cosmological tidal forces."

I think the only reason they calculate that stretching will occur is that they employ a distant observer who is at rest in his own local Hubble flow. Obviously, this observer has a proper recession velocity away from the origin. This motion causes him to perceive a pseudo-time dilation effect in his frame, akin to the classical Doppler effect. But in no way does this indicate that the distance between the particles has stretched in the origin's frame, or in the frame of either test particle. Barnes & Lewis made the oldest mistake in the relativity book, failing to recognize that an apparent effect actually results only from switching their perspective from one reference frame to another.

Jon

Hi Jon, a further reply on your post #3.

On your "another interesting scenario to apply both the 'expanding space' and 'kinematic' models", concluding: "So the result is directionally the same in either model."

I checked: they are also numerically the same and operate as you described in the proper coordinates of the center. I also agree with the collapsing, collisionless dust ball...

However, I think your: "Francis, Barnes et al evidently bollix up the 'expanding space' interpretation of this "outbound" scenario at the end of section 2.6.2 of their paper "Expanding Space: the Root of all Evil?"" may be a wrong interpretation of what they said. Yes, they described two local length measurements in two different inertial frames, but their conclusion is correct, IMO. Cosmic expansion does attempt to stretch things and will do so if things are not kept together by some force.

I compared the difference in peculiar velocities of the two particles when they arrive at comoving distance Χ and when they started. I found that this difference increased, meaning that locally they are drifting apart. This seems to be due to the expansion of space, irrespective of the expansion profile (empty, matter only or matter and Lambda). It is a very slippery issue, though!

It is true that in proper distances from your origin, it appears as if they are coming closer together, but that is hardly a good measurement of what happens to the particles locally. It is somewhat like when you calculate the distance between two particles, originally separated by radial distance Δr, falling radially towards a Schwarzschild hole. Close to the event horizon, one may think that they are approaching each other, because Δr shrinks according to the metric. This is obviously not true where it matters, in local coordinates, because tidal gravity pulls them apart.

-J

Hi Jorrie,

You said:

"It is true that in proper distances from your origin, it appears as if they are coming closer together, but that is hardly a good measurement of what happens to the particles locally."

We're not talking about a BH hole here, with some sort of coordinate singularity "blow up" problem. In a more moderate situation, if the proper distance decreases, then they are in fact coming together, end of story. I'm missing your point.

Jon

Hi again Jon, you wrote: "... if the proper distance decreases, then they are in fact coming together, end of story."

OK, I did misstate the issue for your scenario, which is not quite the same as the one Francis et. al implied. I missed the point that your second particle is fired at a higher local (peculiar) velocity than the local velocity of the first particle at the same time - due to the decreasing local velocity.

Francis stated: "The length of the object is l₀ = v₀∆t₀.", which means the first and last particle are considered to be moving at the same speed relative to the origin at the time the last particle comes past. For the "no internal forces" object, this is valid for small lengths only, as they (indirectly) did assume. Their idea is equivalent to the two-particle setup that I had in reply #15, with the difference that mine also works for large separations, because identical peculiar velocities are ensured.

The fact that particles with identical peculiar velocities move apart like this is hardly surprising. IMO, this is all that Francis et. al tried to show, albeit in a very complicated way!

-J

Hi Jorrie,

For some reason we're wandering around in the swamp here. I think I caused some confusion by describing the convergence of the particles in comoving coordinates as if I were referring to convergence in proper distance coordinates. Regardless, I'm sure that the proper distance between the particles converges over time, they do not separate.

In my scenario, the second particle's peculiar velocity isn't higher because of the decreasing local velocity at the origin. In fact, I specifically said that any tiny decrease in the Hubble flow in the interval between the two firings should be ignored. Just assume that the peculiar velocity at which each particle is fired is identical.

I think the easiest way to visualize this scenario is to consider the distant comoving observer to be the coordinate origin, and set the initial proper firing velocity of each particle such that the firing puts the particle at proper rest (0 proper velocity) relative to the origin. Now we've just transformed this from an "outbound" scenario to an "inbound" scenario, and each particle behaves as I described for the inbound scenario. As the cosmic Hubble rate decays, the particles will approach the origin, and the proper distance between the particles will decrease.

If both particles had been fired simultaneously with the same initial proper velocity (zero) toward the origin from cannons located at different radial proper distances from the origin, the two particles would converge exactly as they reach the origin, at which point the second particle would pass the first particle.

However, our scenario is different because the particles are fired from different radial proper distances from the origin and at different times. As a result, the arrival of the second particle at the origin is time-delayed compared to the arrival of the first particle. This additional time delay does not reflect any increase in proper distance between the particles after the second particle is fired. The entire time delay (and resulting increase in proper distance) occurs before the second particle is fired.

Francis & Barnes talk about the pre-firing time delay as if it occurred post-firing. Tsk, tsk. The two "outbound" particles do not separate at all during the second particle's journey, and clearly 'expanding space' does not cause the "object" to stretch, when one considers only the time period when the entire "object" exists.

It is very easy to get fooled by the 'expanding space' paradigm into predicting that the two particles will progressively separate. That's why it's really helpful to have the kinematic alternative mechanism available to quickly double-check the predictions.

I believe that in the scenario as I described it, the proper distance between the particles will decrease, but the second particle never will pass the first particle. Their proper locations will asymptotically converge at infinity.

Jon

Hi Jon, me again.

The "swamp" that you referred to is caused by subtle differences in initial conditions, as you also remarked later. You are right for your scenario, but I think Francis et. al are right for theirs as well (because I think my scenario is equivalent to theirs and my particles separate, as I have checked with a spreadsheet).

You created a variant of the "2 tethered galaxies" experiment, as your "inbound" scenario illustrates. Francis et. al created a hypothetical object that is virtually impossible to shoot from a single location. In the style of Born-rigid acceleration of a rod in SR, they have to shoot each particle at a constant peculiar velocity for its location and all at the same cosmic time. The last particle must actually be shot at a lower proper velocity than the first, because it is at negative proper distance, but this is very poorly defined by them. They just talk about a velocity for the particle.

However, cosmic tidal stretching can't be argued away by their poor definition. It is not necessary to accelerate things to demonstrate. Let a number of lined-up comoving observers 'build' the hypothetical object by placing particles at rest with themselves. The whole object will remain at rest in the CMB frame (zero peculiar motion for all particles), but the object will obviously 'stretch' with the expansion factor of an expanding cosmos, irrespective of the expansion law.

An interesting variant of this is: after some time, simultaneously place rigid massless rods between adjacent particles. The center particle will obviously jerk all the rest 'inward' in order to maintain a constant proper distance from the center, i.e., the rods will experience a stretching shock-stress. After that, the stresses in the rods will depend on the expansion law. In the 1,0 case, the stresses will become a squeeze and in the 0.3,0.7 case it will become a stretching force. Either way, there are cosmic tidal stresses on rigid objects (and separations or convergence of free particles), depending on the situation defined. I believe this is also true inside gravitationally bound objects, but just too feeble to observe.

-J

Hi Jorrie, you said "they have to shoot each particle at a constant peculiar velocity for its location and all at the same cosmic time. The last particle must actually be shot at a lower proper velocity than the first because it is at negative proper distance..."

I interpret the Frances & Barnes scenario to be different and simpler than you do. They say:

"It [the object] is shot a way from the origin (χ = 0) with speed v₀, the first particle leaving at time t₀ and the last at t₀ + Δt₀. The length of the object is l₀ = v₀Δt₀."

It seems clear that the first and last particle are not shot at the same cosmic time, and they are each shot out with equal peculiar velocity (v₀), and equal proper velocity relative to the origin and each other.

When I "converted" the scenario to an inbound one, I didn't intend to change any of the dynamics of motion, only to make it easier to recognize that the particles behave much like a standard untethered galaxy.

Jon

Hi Jon, you wrote: "It seems clear that the first and last particle are not shot at the same cosmic time, and they are each shot out with equal peculiar velocity (v₀), and equal proper velocity relative to the origin and each other."

I guess the only way to really know is to ask the authors what they meant, but here are a few more reasons why I think your assumption (which correctly produces your result) is not compatible with Francis et. al (2007):

• They do not say what the velocity of any particle is at any time, just that the object is shot away at v₀, in such a way the lead particle leaves (the origin) at t₀ and the last particles leaves it at t₀ + ∆t₀.
• The only way I know of that is compatible with their result is for each particle to have the same peculiar velocity at the same cosmic time.
• I could find no problems in their math of 2006 and 2007, just that they described the (2007) initial conditions inadequately.
• Finally, Barnes et. al (2006) is the best generalized analysis on expanding space that I've seen so far. I would seriously doubt that they made a fundamental error in the 2007 paper, using results from the former, without peers having discovered it up to now.

-J

Hi Jorrie,

Well, we can just agree to disagree.

If there is a Δt for the last particle, then that precludes your suggestion that the particles were all shot at the same cosmic time. There is no ambiguity there.

I actually corresponded with Wallace about the subject some months ago. You probably know he is one of the authors of the article. I wasn't as articulate in explaining the problem then as I am now. He justified the confusing conclusion in that section by arguing that Doppler-like stretching is close enough to being equivalent to actual stretching. I disagreed then and I disagree now.

We all make mistakes. You have corrected some of Wallace's mistaken statements on Physics Forums in the past, just as I suppose he corrected some of yours... and lots of mine.

Jon

Hi Jon, OK, I'll give it one last try, before we decide to agree or disagree.

In order for the Francis et. al thought experiment to be able to distinguish between internal forces (holding the object together) and cosmological forces (trying to alter its shape), there must be no other forces than cosmological ones working on individual particles (after t₀, that is).

The simplest way to do that is what I've described in #15 and #21, but since we can argue about their equivalence with the Francis scenario, here's another one that is 'more equivalent' to their scenario.

Let the 'no internal forces' object, consisting out of many co-linear particles, fly past you (at the origin) at time t₀. You measure the speed of the first particle (v₀₎, using Doppler when it passes you at zero range. At that same cosmological time (t₀), I happen to be cosmologically comoving at proper distance -L₀ from you and I measure the speed of the last particle to also be v₀. You take the time when the last particle passes you as t₀+Δt₀. We confer and conclude that at time t₀ the 'whole object' was moving at v₀ and it had a proper length L₀=v₀/Δt₀.^[1] By its definition, proper distance can only be directly determined by measurements at the same cosmological time (the same as the simultaneity and proper length issue in SR).

In this experimental setup, the two particles have identical peculiar velocities at time t₀. Their peculiar velocities can be shown to remain identical, with both declining the same over cosmological time as they join the Hubble flow. The rest is easy - with identical peculiar motions, the proper length of this 'no internal forces' object can only increase over cosmological time (given that there is expansion and not contraction).

I rest my case.

-J

[1] In any expanding universe, the last particle will have a smaller peculiar velocity than v₀ when it passes the origin (at t₀+Δt₀) - you cannot ignore the fact that peculiar velocities decay with time. Your scenario has identical v₀ at different times and hence different peculiar velocities at the same time. It is obviously not the same, because different peculiar velocities diverge over time.

Hi Jorrie,

Thanks for the further explanation. Unfortunately, we still disagree about which of the two scenarios Francis & Barnes actually intended.

Again, I think it is clear they intended for each particle to be shot away from the same origin location, at different cosmic times. They specifically say: "... the first particle leaving at time t₀ and the last at t₀ + Δt₀. " This categorically rules out any scenario where all of the particles "leave" at the same cosmic time. Conversely, if they had intended the scenario you describe they might have used the word "arriving" instead of "leaving", or more likely they would have phrased the statement quite differently.

In any event, the particle behavior is a simple consequence of whatever particular initial conditions we select for the scenario. Consider the "outbound" version of my and your scenarios:

(a) If we define the initial conditions such that each particle in a string begins with exactly the same proper velocity relative to the origin, then the proper distance between the particles will subsequently shrink.

(b) If, however, we define the initial conditions such that each particle begins with exactly the same peculiar velocity relative to its own local Hubble flow, then of course the rear particle will have a lower proper velocity (relative to the origin) compared to the front particle, and the proper distance between the two particles will subsequently stretch in direct concordance with the expansion of the scale factor.

In the latter case, it doesn't add much to claim that the stretching is due to an expansion of the hypersphere. It is simpler to state the obvious: the stretching is due to the fact that we specified initial conditions such that the particles began in a state of proper motion relative to each other.

. . . . . . . . . .

The main reason I asked you to disregard the fact that the proper velocity of the first particle decreases by the time the second particle is shot away is that it just complicates the picture. The second particle does not require a higher starting proper velocity in order to catch the first particle; they will eventually converge even if the second particle begins with zero proper velocity relative to the first particle.

In any event, I believe that the change in the first particle's velocity by the time the second particle shoots always causes the second particle to gain on the first particle more quickly than it would have in the absence of such a velocity change. Let's separately consider the "outbound" and "inbound" scenarios.

Outbound. In the outbound scenario, both particles are shot away from the origin at the same initial proper velocity. However, the proper velocity of the first particle (relative to the origin) decelerates slightly by the time the second particle shoots. This allows the second particle to gain on the first particle more quickly than it would have if their relative proper velocities were still zero when the second particle launched.

In terms of peculiar velocities, by the shooting time of the second particle, the first particle's peculiar velocity has decayed by 1/a. The second particle starts with a peculiar velocity slightly higher than the first particle's, again indicating that the second particle will gain on the first particle more quickly than if their peculiar velocities were still the same when the second particle launched.

Inbound. In the inbound scenario, both particles again are shot toward the origin at the same initial proper velocity. However, the proper velocity of the first particle accelerates slightly toward the origin by the time the second particle shoots. This acts to increase the first particle's proper velocity relative to the second particle. But at the same time, the proper recession velocity of the cannon (relative to the origin) has also decreased, which acts to increase the second particle's proper velocity relative to the first particle. The kinematic model tells us that the cannon's recession velocity decelerates more than the first particle's proper velocity accelerates, because in this scenario the cannon's "gravitational sphere" at each instant in time after the first particle's shooting is always larger than the first particle's gravitational sphere. Meaning that the second particle again will approach the first particle more quickly than it would have if their proper velocity relative to each other had been zero at the second particle's shooting time.

In terms of peculiar velocities, by the shooting time of the second particle, the first particle's peculiar velocity has decayed by 1/a, same as in the outbound scenario. The second particle again starts with a peculiar velocity slightly higher than the first particle's, indicating that the second particle will gain on the first particle more quickly than if their peculiar velocities were still identical when the second particle launched.

Jon

Hi Jon, you wrote: "Thanks for the further explanation. Unfortunately, we still disagree about which of the two scenarios Francis & Barnes actually intended."

Since I've delivered my closing arguments and I take it that post #29 is your closing arguments, we will have to wait until some Judge or Jury delivers a verdict, I guess!

-J

Hi Jorrie,

Fine, but do you agree with my comparison in post #29 of your version against my version of the scenario?

Jon

Hi Jon, on your question: "Fine, but do you agree with my comparison in post #29 of your version against my version of the scenario?"

Yes, but then recall the argument in #27 that my version of Francis et. al is the only one where a velocity for the particle and its proper length make any sense, i.e.:

"We confer and conclude that at time t₀ the 'whole object' was moving at v₀ and it had a proper length L₀=v₀/Δt₀. By its definition, proper distance can only be directly determined by measurements at the same cosmological time (the same as the simultaneity and proper length issue in SR)."

I still rest...

-J

Hi Jorrie,

I am more interested in the substantive behavior of the scenario I call (a) because, like the tethered galaxy problem, it contradicts some aspects of 'expanding space' intuition. The scenario I call (b) isn't interesting, because it's intuitively obvious that two particles which begin the exercise receding away in concert with the Hubble flow will continue thereafter to recede away from each other in concert with the Hubble flow. It also defies common sense to characterize a collection of particles as an "object" if they do not start out at rest relative to each other. Scenario (b) tells us nothing about how a real object (even one lacking internal electromagnetic forces) will behave in an expanding universe, so if Scenario (b) is what Francis & Barnes intend, their analogy is still a complete failure.

I do not think Francis & Barnes wanted to complicate their analogy with your technical concern about measuring the instantaneous proper length of the "object". They say "If we assume that Δt₀ and Δt_f are small, it follows that we can assume C₀ = C_f = C..." They seem to just assume your concern away.

Jon

Hi Jon, we are going in circles again...

Recall my #24, and later #33, where I have shown (maybe loosely) that your own scenario also ends up with the particles separating, once they have crossed over, after billions of years...

Bottom line is, whether Francis & Barnes were right in their statements of initial conditions or not, you still cannot argue 'cosmic tidal forces' away. Maybe it's hard to see them in the kinematics model, but they are still there.

-J

Hi Jorrie,

Well we should probably wrap up this discussion. It is pointless to debate whether the fact that an "object" without internal forces will immediately collapse and turn inside out is more or less relevant than the fact that the inside-out-transformed object will eventually begin stretching billions of years later. Both are relevant characteristics. But one cannot give credit to Francis & Barnes for highlighting the post-collapse stretching phase in their section 2.6.2 scenario -- obviously that was not the focus of their analogy.

I find the kinematic analysis to be easier to visualize quickly and less likely to lead one down the wrong intuitive path. Both the collapse phase and the post-collapse stretching phase are particularly easy to visualize kinematically. But at the end of the day, the kinematic analysis is no more or less valid than the 'expanding space' analysis, so its entirely a matter of personal preference.

Jon

Hi again Jon, I agree.

I'm starting a new Blog post soon, specifically aimed at the balloon analogy, much like Marcus' thread on PF, just with a few engineering twists!

Once that one has run its course, maybe we can do one on the kinematic model as well, drawing on your preference for and experience with it.

-J

Hi again Jon. Some problems with your analysis of the "no internal forces object":

In #19 you wrote: "In my scenario, the second particle's peculiar velocity isn't higher because of the decreasing local velocity at the origin. In fact, I specifically said that any tiny decrease in the Hubble flow in the interval between the two firings should be ignored. Just assume that the peculiar velocity at which each particle is fired is identical."

This means that at the time of your 2nd particle's firing, its proper velocity is higher than the proper velocity of the 1st particle, which has already been decelerated. You cannot simply ignore the kinematics (or the decreasing Hubble rate) now. With your initial conditions, yes, they will approach each other, but at an increasing relative proper speed.

You closed #19 with: "I believe that in the scenario as I described it, the proper distance between the particles will decrease, but the second particle never will pass the first particle. Their proper locations will asymptotically converge at infinity."

I assume that you have already realized that your intuition is incorrect here. The particles will pass each other and then asymptotically join the Hubble flow for their reversed positions in the (1,0) universe. Your scenario just delays the 'cosmic tidal stretching' by a few billion years!

-J

Hi Jorrie,

I haven't realized that my intuition was wrong, but I would appreciate if you can show why it is wrong. I don't have a preference in one result over the other, I just want to get to the correct answer.

Jon

Hi again Jon, on the 'intuition issue': I've also stumbled over this one in the past, but I'll try it in kinematics. For simplicity, consider simply escape velocity and not 'hyper-escape velocity', as the scenario would require (escape velocity here would be equivalent to zero peculiar velocity).

In a static 'star interior' kinematic solution, two particles shot at different times, each at escape velocity, from the origin will asymptotically converge at 'infinity', as 'standard intuition' confirms.

However, in an expanding (density decreasing) interior solution, the second particle overtakes the first one, if shot at the same proper velocity at a later time, because then the second particle has been shot at higher than escape velocity in the less-dense star.

As fully described in #27, my interpretation of the Francis scenario is that the second particle is effectively shot at a lower proper velocity than the first one, to such an extent that their proper separation increase ad infinitum. A kinematic spreadsheet confirms this, provided that you get the initial conditions exactly right (ensuring identical peculiar velocities at the same time), which is not that easy in the straight kinematic model. It is much easier in the hyper-spherical model...

-J

Hi Jorrie,

I think your conclusion here just reflects a scenario with different initial conditions than I intended when I gave my intuitive conclusion. In the terminology of my post #29 today, I'm talking about scenario (a) and you're talking about scenario (b).

I'd like to figure out whether in scenario (a) the particles will converge at infinity, or whether the second particle instead will pass the first particle after some finite elapsed time.

Jon

Hi again Jon, at least the 'intuition case' is still open.

You said: "I'd like to figure out whether in scenario (a) the particles will converge at infinity, or whether the second particle instead will pass the first particle after some finite elapsed time."

I did describe your scenario (a) when I wrote in #29: "However, in an expanding (density decreasing) interior solution, the second particle overtakes the first one, if shot at the same proper velocity at a later time, because then the second particle has been shot at higher than escape velocity in the less-dense star." (Emphasis added).

I have simulated this and found the particles to pass each other. Maybe you should do the same to confirm or refute my conclusion.

Remember, that it is an 'interior to matter' scenario, not an 'exterior' solution, since you have to constantly change the amount of matter 'inside' the particle's radius - which is the same thing as constantly changing the density on the surface of the hyper-sphere...

-J

Hi Jorrie,

OK, on further consideration, your #28 does appear to describe my version of the scenario (a), i.e. that both particles have identical proper velocities at the instant the second particle is shot off. So I'll agree with you that in my scenario (a) the second particle will eventually pass the first particle.

Jon

Hi Jon, just an update on my reply #11, on the "Root of all Evil" ArXiv paper. I wrote: "I compared the difference in peculiar velocities of the two particles when they arrive at comoving distance Χ and when they started. I found that this difference increased, ..."

Not a good way to try and determine the effect; neither is the way that Francis, Barnes et. al used a very good one.

The easiest way I found to explain it is by lining up three comoving cosmological observers A, B and C, with distances AB and BC equal. At some predetermined cosmological time t₀, A shoots a particle p_A towards B and B shoots a particle p_B towards C, both at identical local speeds. The initial proper distance between the two particles is obviously the proper distance between A and B at t₀. By the cosmological principle, particle p_A will pass B at the same cosmological time t₁ as when p_B passes C. Now the proper distance between the particles will be the proper distance between B and C. I'm sure you already see what I'm getting at.

If there was any shape or form of cosmic expansion, the proper distance between B and C at time t₁ is obviously larger than what the proper distance between A and B was at time t₀. Hence, the particles separated in proper distance - the 'cosmological tidal force' of Francis, Barnes et. al.

Is this perhaps another case where the kinematic model runs into trouble?

-J

Hi Jorrie,

Great chart and explanations. This goes a long way to making me feel like I am getting caught up with your 'debate'. I think you and Jon have made a great accomplishment in showing that the 2 models are equivalent in a (1,0) universe. However, since that is not the presently accepted model, I think the hot topic argument is not over yet. After reading Jon's remarks, I'm glad I didn't read that paper. I might have been 'lead down the garden path'. It's great to have the two of you with such good insight and mathematical prowess to expose the fallacies of people who are supposedly 'in the know'. Keep up the good work.

Regards

-S

Hi S, you wrote:

"I think you and Jon have made a great accomplishment in showing that the 2 models are equivalent in a (1,0) universe. However, since that is not the presently accepted model, I think the hot topic argument is not over yet."

Yes, it has been great to have a knowledgeable, good debater like Jon participating and in the process we have all learned something. I think the (0.3, 0.7) universe has now also been shown equivalent for the two views. We are still debating the cosmological time issue - Jon must still convince me that his favorite model produces uniform cosmological time for particles 'moving with' the Hubble flow.

AFAIK, the hyperspherical model does not suffer from the issues that I mentioned in reply #8. All particles on the sphere are at the same gravitational potential and when they have no peculiar movement, they all record the same cosmological time. There is no need to find a velocity time dilation to cancel out a gravitational time dilation.

-J

Could you explain to me in idiot speak?

Hi Epke, your request reminds me about a previous colleague (a mechanical engineer) that wrote an internal paper titled: "Aerodynamics for Electronic Engineers and Other Idiots."

Needless to say it did not go down too well in a Company with mainly electronics guys...

What Jon and myself are discussing is not quite rocket science, although it touches the 'bleeding edge' of cosmology. I suggest you take a look at the index of this Blog (in the header) and read some previous cosmology posts - then ask again!

-J