The Mystery of Superluminal Recession Velocities in Cosmology

[marcus]

It seems to me that what we are talking about here is an unusual coordinate system that very few cosmologists use. Milne was an early proponent---he didnt like the idea of space expanding and proposed the Milne universe, I think that was back in 1930s or 1940s.

Lately Chodorowski seems to be the main proponent. I think of him as marginal because almost nobody cites his papers-----single digit citation of his papers since 2000, I will list them.

As I understand it, Chodorowski challenged the idea of superluminal expansion. He used his maverick coordinate system and argued that it didn't happen. Then Lewis et al (LFBJ) REFUTED Chodorowski's argument and showed that even in his conformal coordinates distances were increasing at rates greater than c.

Here are Chodorowski's papers since 2000. He has authored 11, it appears.
http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+A+CHODOROWSKI+AND+DATE+%3E2000&FORMAT=www&SEQUENCE=citecount%28d%29
The first 6 papers have been cited.

1) A direct consequence of the expansion of space?
Michal Chodorowski (Warsaw, Copernicus Astron. Ctr.) . Oct 2006. 8pp.
Published in Mon.Not.Roy.Astron.Soc.378:239-244,2007.
e-Print: astro-ph/0610590
Cited 6 times

2) Cosmology under Milne's shadow.
Michal J. Chodorowski (Warsaw, Copernicus Astron. Ctr.) . Mar 2005. 12pp.
Published in Publ.Astron.Soc.Austral.22:287,2005.
e-Print: astro-ph/0503690
Cited 5 times

3) Is space really expanding? a counterexample.
Michal Chodorowski (Warsaw, Copernicus Astron. Ctr.) . Jan 2006. 10pp.
Published in Concepts Phys.4:17-34,2007.
e-Print: astro-ph/0601171
Cited 4 times

4) Likelihood analysis of the Local Group acceleration revisited.
Pawel Ciecielag (Munich U., Inst. Astron. Astrophys. & Warsaw, Copernicus Astron. Ctr.) , Michal J. Chodorowski (Warsaw, Copernicus Astron. Ctr.) . Jan 2004. 7pp.
Published in Mon.Not.Roy.Astron.Soc.349:945,2004.
e-Print: astro-ph/0401195
Cited 3 times

5) Local group velocity versus gravity: the coherence function.
Michal Chodorowski, Pawel Ciecielag . Sep 2001. 9pp.
Published in Mon.Not.Roy.Astron.Soc.331:133,2002.
e-Print: astro-ph/0109291
Cited 2 times

6) Superluminal apparent motions in distant radio sources.
Michal J. Chodorowski (Warsaw, Copernicus Astron. Ctr.) . Jul 2004. 4pp.
Published in Am.J.Phys.73:639-643,2005.
e-Print: astro-ph/0407478
Cited 2 times

7) Local group velocity versus gravity: nonlinear effects.
P. Ciecielag, M. Chodorowski, A. Kudlicki . Jan 2001. 15pp.
e-Print: astro-ph/0101078

8) New era in likelihood analyses of the local group acceleration.
Michal Chodorowski, Pawel Ciecielag . Feb 2002. 2pp.
e-Print: astro-ph/0202296

9) Precision analysis of the Local Group acceleration.
Michal Chodorowski, Pawel Ciecielag (Warsaw, Copernicus Astron. Ctr.) . Oct 2003.
e-Print: astro-ph/0310895

10) The optimal window for the 2MASS dipole.
Michal Chodorowski, Jean-Baptiste Coiffard, Pawel Ciecielag, Stephane Colombi . Jun 2007. 9pp. Temporary entry
e-Print: arXiv:0706.0619 [astro-ph]

11) Cosmic velocity--gravity relation in redshift space.
Stephane Colombi, Michal Chodorowski, Romain Teyssier . May 2008. 24pp.
Temporary entry
Published in MNRAS, 375, 2007, 348.
e-Print: arXiv:0805.1693 [astro-ph]

Well company has come! Have to go. Let's everybody respect Michal! I'm the last to denigrate clever mavericks. But he is way out in left field, keeping the Milne approach alive and arguing against superluminal expansion. Even though they debunked his most noticeable conclusion, LFBJ are actually helping Chodorowski in a sense because at least they cite his research----and most of his citations probably are self-cites. He needs all the visibility he can get!

But IMO the main moral here is that if you invoke peculiar coordinate systems you should be very clear and explicit about the math consequences. Verbal interpretation is not going to be straightforward. Like the spatial hypersurfaces are going to be totally different. What does it mean to all do something at the same time etc etc.

Anyway, company here. Have to get back to this later.

 

[jonmtkisco]

Hi Marcus,

It seems to me that what we are talking about here is an unusual coordinate system that very few cosmologists use. Milne was an early proponent---he didnt like the idea of space expanding and proposed the Milne universe, I think that was back in 1930s or 1940s.

Good grief. Now you are piling abuse on the whole concept of conformal coordinates, seemingly without trying much to understand them.

Chodorowski does NOT use the Milne model, because as he points out, Milne designed it specifically without GR and intended it as a rejection of GR; it is based only on SR. That's why it is not useful except in an empty universe.

The conformal coordinates Chodorowski and Lewis & Francis use comply fully with GR and take gravity into account. They are not "weird" and they were not invented by those authors. They were developed by Infeld & Schild in 1945 and further developed by Landau & Lifgarbagez in 1979.

Regarding use of conformal representations of FLRW universes to consider radial motion, Lewis & Francis say: "Such an approach has proved to be very powerful in understanding cosmic causality and the nature of fundamental horizons in the Universe (Rindler, 1956; Ellis & Stoeger, 1988)."

Also, please note that my question about superluminal recession does not depend at all on using conformal coordinates, or any particular coordinate system. I suggested in one post that given how powerful conformal coordinates are, perhaps they could be useful in cutting through the fog here. I agree that when one actually uses a particular coordinate system, one should apply it carefully, including clock effects.

Jon

 

[yuiop]

Have a look at the attached diagram which is a slightly modified version of the one Ned Wright's excellent Cosmology website http://www.astro.ucla.edu/~wright/sne_cosmology.html

The vertical and horizontal purple lines indicate the best match of SN1a supernova data, CMB data and density estimates from observations of clusters which seems to point towards a cosmological constant or Omega(Lambda) value of 0.701 and a Omega(Mass) giving an Omega(total) value of around 1.01. Of course error margins do not exclude the possibility of a total that is exactly 1.0 which a lot of people would like.

My concern is that if the supernova data is considered alone, it points to a much higher cosmologial constant and Omega(mass) value as indicated by the vertical and horizontal green lines. Why is the green dot at the centre of the supernova data error ellipses so far off the data from the CMB observations? It seems unlikely that sytematic errors are the cause of this and seems more likely that some fundamental assumption is used when analysing the supernova data and that when this false assumption is identified and corrected, it will bring the supernova data into line with the CMB and cluster data.

Ned Wright points out that another possible factor is the "the equation of state parameter W". The effect of adjusting that parameter is shown in a multipanel diagram right at the bottom of the link I gave above. Even adjusting that parameter does not bring the supernova data into line with the rest of the data. Something is amiss.

 

[Chaos' lil bro Order]

It is more like a succession of progressively faster conveyor belts, so there may be a limit to how much of a head start FFA has, that will still allow you to catch up with it. Maybe someone has a formula for that?

P.P.S. The headstart limit is probably the Hubble Horizon.

If you travel at 0.9c you will never reach a galaxy at 2.0c, period. You cannot get around this fact in a flat accelerating universe.

 

[Chaos' lil bro Order]

At some time in the past, we have sent a message to a galaxy (which then was closer to us but is NOW at twice Hubble distance or about 27 billion LY and receding at speed 2c) asking them to launch a spacehip towards us at speed 2c.

That is what would be necessary, to achieve the key condition Jon requires. If they could do that, then the ship would have zero proper speed with respect to earth. (Zero proper speed relative Earth is what Jon is asking for.)

If you use the simplified, red, version of Jon's proposal, then as far as I can see it gets rid of the problems with coordination that you mentioned.

I see your point Marcus. We then run into a tiny problem of launching a spaceship at 2.0c. So whether we are communicating at 2.0c or moving at 2.0c, the problem is the same, its impossible given our current physics models

 

[jonmtkisco]

Hi Chaos,

If you travel at 0.9c you will never reach a galaxy at 2.0c, period. You cannot get around this fact in a flat accelerating universe.

I thought we were talking about a non-accelerating universe. Anyway, consider this passage from Francis & Barnes et al, Expanding Space: the Root of all Evil (7/07):

While the picture of expanding space possesses distant observers who are moving superluminally, it is important not to let classical commonsense guide your intuition. This would suggest that if you fired a photon at this distant observer, it could never catch up, but integration of the geodesic equations can reveal otherwise (this is very clear in the conformal representation of FRW universes (Chodorowski 2006b);...

Davis & Lineweaver, "Expanding Confusion: common misconceptions of cosmological horizons and the superluminal expansion of the universe" (11/03) has an entire section devoted to this point, summarized as follows:

Light that superluminally receding objects emit propagates towards us with a local peculiar velocity of c, but since the recession velocity at that distance is greater than c, the total velocity of the light is away from us (Eq. 20). However, since the radius of the Hubble sphere increases with time, some photons that were initially in a superluminally receding region later find themselves in a subluminally receding region. They can therefore approach us and eventually reach us. The objects that emitted the photons however, have moved to larger distances and so are still receding superluminally. Thus we can observe objects that are receding faster than the speed of light.

They say this is true both in a decelerating and accelerating universe.

Jon

 

[Haelfix]

Its true in any universe. When people measured the recession speed of some quasars early on, they were shocked to see speeds > c.

It caused endless amounts of confusion until it became clear that it was a simple purely classical optical trick, and had nothing to do perse with whether or not we are in an open, closed or critical universe.

Anyway, as for the papers under discussion. I briefed through it, and don't really understand the point or where the new physics is. Its just coordinate semantics and interpretational mumbo jumbo, and I don't see any observable that's measurable (even in principle) to give different results than the standard formulation. Indeed the authors seem to agree.

A rule of thumb in GR. Whenever there is confusion or ambiguity about whether such and such an effect is a gauge artifact or a 'real' physical property, you always must look at relationships between physical observables perse (and be careful b/c they are often redefined subtly under transformations).

 

[jonmtkisco]

Anyway, as for the papers under discussion. I briefed through it, and don't really understand the point or where the new physics is.

Hi Haelfix, I'm not sure which papers specifically you're referring to. As far as I can tell, the two papers I cited do not purport to invent new physics, they attempt to provide a lucid explanation for interesting aspects of standard cosmology.

In my opinion, the field of cosmology would benefit greatly from more detailed explanation and less dismissive elitism. In general, I find the standard textbooks in this field to be atrocious in their lack of clarity. Not just because they waste too much ink on pedantic proofs of equations (which they do), but more importantly because they provide too little clear textual interpretation. So any paper that adds useful examples and explanation is welcome for the 95% of us sad sacks who aren't members of the elite.

Jon

 

[Haelfix]

Hi Jon.

Thats perfectly fine, I am all for people getting a better grasp of the physics with examples and so forth. However in my experience messing around with too many nonstandard coordinate transformations typically confuses the issue rather than elucidating intution. On the one hand, people might not know what you are talking about, and interpret a perfectly valid statement erroneously. On the other it doesn't necessarily simplify calculations either.

Historically there are many famous examples of this happening. Einstein for instance was deeply confused about gravitational waves, and whether or not they were physical or merely a gauge artifact (he thought the latter for awhile). Recently, see the controversy over Unruh radiation and Rindler coordinates, or the VSL theories.

Sometimes the issues can be very technically subtle yet stem from something as naively benign as a coordinate transformation.

 

[jonmtkisco]

However in my experience messing around with too many nonstandard coordinate transformations typically confuses the issue rather than elucidating intution.

Yes I agree Haelfix. A lot of the debate among the recent technical papers on expanding space seems to involve each author pointing out an inaccurate coordinate transformation made by a prior author. Clock synchronization seems to be the Achilles Heel.

However, I think these authors do the cosmology community a great service when they take the risk of laying out a "physical" conclusion discerned through coordinate transformations. Even when their conclusion is wrong (which happens to everyone on occasion), the papers stimulate other cosmologists to come out of hiding and publish a correction and revised explanation. This public process of hypothesis, analysis, correction, and revised hypothesis is the best way for the community to make progress in my opinion. No single author is going to figure it all out alone, at least not in our lifetimes.

On the other hand, cosmologists who are too timid to publicly venture even one conceptual step away from exact solutions to the Einstein equations are going to contribute much less to advancing the state of community knowledge.

That's not to say that cosmology needs more "crackpots." If people aren't willing to admit to their errors or sloppy explanations, and learn from them, then they are more of a hindrance than a help to the community.

Jon

 

[Chaos' lil bro Order]

Jon

Maybe I am just tired, but I don't understand how this occurs:

'Light that superluminally receding objects emit propagates towards us with a local peculiar velocity of c, but since the recession velocity at that distance is greater than c, the total velocity of the light is away from us (Eq. 20). However, since the radius of the Hubble sphere increases with time, some photons that were initially in a superluminally receding region later find themselves in a subluminally receding region. They can therefore approach us and eventually reach us. The objects that emitted the photons however, have moved to larger distances and so are still receding superluminally. Thus we can observe objects that are receding faster than the speed of light.'

I can't follow the logic of this statement that allows for an object that was in a superluminally receding region, to all of a sudden be in a subluminal receding region, simply because the Hubble sphere is increasing in radius over time. Can you explain this to me please.

Cheers.

 

[jonmtkisco]

Hi Chaos,
This concept is non-intuitive. I recommend you read the Davis & Lineweaver http://arxiv.org/abs/astro-ph/0310808" I cited, and then come back with specific questions. I can answer only pretty easy questions about it!

Jon

 

[yuiop]

Jon

Maybe I am just tired, but I don't understand how this occurs:

'Light that superluminally receding objects emit propagates towards us with a local peculiar velocity of c, but since the recession velocity at that distance is greater than c, the total velocity of the light is away from us (Eq. 20). However, since the radius of the Hubble sphere increases with time, some photons that were initially in a superluminally receding region later find themselves in a subluminally receding region. They can therefore approach us and eventually reach us. The objects that emitted the photons however, have moved to larger distances and so are still receding superluminally. Thus we can observe objects that are receding faster than the speed of light.'

I can't follow the logic of this statement that allows for an object that was in a superluminally receding region, to all of a sudden be in a subluminal receding region, simply because the Hubble sphere is increasing in radius over time. Can you explain this to me please.

Cheers.

Take at look at this diagram from Ned Wright's cosmology website http://www.astro.ucla.edu/~wright/omega0.gif.

The curved red line forming the tear drop shape is the path of a photon from a superluminal object back to Earth. The straight lines with small light cones superimposed on them are the worldlines of receding galaxies. The straight lines tilted more than 45 degrees from the horizontal represent superluminal galaxies beyond the Hubble sphere. As long as the line remain straight they remain beyond the expandng Hubble sphere which lies on the null 45 degree lines. Light or any object moving with peculiar motion directed towards Earth relative to the superluminal galaxies follow a curved line that eventually crosses the Hubble sphere. The red line clearly shows the photon initially receding away from Earth and eventually turning around and start heading towards Earth. If you look at where the photon path intersects the Hubble null line you will see the photon worldline is briefly vertical so the the photon is neither gaining or losing ground but just breaking even and reamaining at constant distance from Earth but the expnanding Hubble horizon puts the the photon into the sub luminal zone where it starts to make headway. A subluminal particle with rest mass would still be losing ground at the Hubble horizon but eventually the expansion of the horizon puts it in a zone where it gains ground. I hope you can see it is not just simply because the expanding Hubble radius is increasing over time. The particle has to have some local motion directed towards the Earth in order to eventually be included inside our Hubble sphere. The same basic principle applies in an accelerating or a matter dominated decelerating universe. The Hubble radius is no longer a straight null line in those cases but a curved line that accelerates or decelerates along with the rest of the universe. The Hubble horizon is not a barrier to light or particles moving across the horizon in wither direction so it is not like the event horizon of a black hole and in fact has no real physical significance which shows it is artifact of using co-moving coordinates and there is no real significance to the demarcation between subluminal and superluminal velocities that appear as a result of using the comoving coordinate system. In conformal coordinates all visible galaxies are subluminal. Hope that helps :)

 

[yuiop]

Marcus,

I agree that the general consensus on this forum is that space does not expand, only distances. I remain unconvinced however, since the concept of space (or spacetime) expanding so simply explains superluminal recession velocities. Not to mention that some notable cosmologists seem to buy into the idea of expanding space. To quote Edward Harrison:

"The answer is that galaxies are not moving through space but are moving apart by the
expansion of intergalactic space...

...recession is a result of the expansion of space that obeys the rules of general
relativity, and is not like motion through space that obeys the rules of special relativity.

Those persons who find it difficult to understand that recession is without limit usually
make the mistake of thinking that the receding galaxies are like projectiles shooting away
through space. This is an incorrect view. The correct view is of galaxies more or less at
rest in expanding space."

Edward Harrison, "Cosmology The Science of the Universe", 2nd edition, 2000, page 282

Is Harrison misinformed, or has there been a breakthrough in understanding since he wrote this? He seems pretty darned clear about his view of expansion of space.

The trouble is that cosmologists that dismiss the "projectile interpretation" hardly ever seem to to show what that model model predicts and how that is clearly contradicted by acual observations. It comes across that they object purely on philosophical grounds. Tomorrow when I have more time I hope I can demonstrate a clear difference in the predictions of the conformal (projectile) model and the co-moving (expanding space) models and further demonstrate that an actual observation is clearly in favour of the former and rejects the latter.

 

[Wallace]

The trouble is that cosmologists that dismiss the "projectile interpretation" hardly ever seem to to show what that model model predicts and how that is clearly contradicted by acual observations. It comes across that they object purely on philosophical grounds. Tomorrow when I have more time I hope I can demonstrate a clear difference in the predictions of the conformal (projectile) model and the co-moving (expanding space) models and further demonstrate that an actual observation is clearly in favour of the former and rejects the latter.

I'd be interested to see if you can do this. As far as I can see every time this is looked at seriously the conclusion is resoundingly clear, both approaches (expanding space or galaxies flying apart) are exactly equivalent and every suggestion of an observable difference has at heart some mathematical error. When the co-ordinates are dealt with correctly observables are unchanged, as they should be.

The only contrast between the approaches is a question of which might be the best to guide intuition. When the handle is properly cranked it is clear that they two approaches are just different co-ordinate descriptions of the same physics. If you think you can show otherwise I'd be interested to see your argument, since I can't imagine how this would be possible myself.

 

[yuiop]

Special Relativity and cosmology

The trouble is that cosmologists that dismiss the "projectile interpretation" hardly ever seem to to show what that model predicts and how that is clearly contradicted by actual observations. It comes across that they object purely on philosophical grounds. Tomorrow when I have more time I hope I can demonstrate a clear difference in the predictions of the conformal (projectile) model and the co-moving (expanding space) models and further demonstrate that an actual observation is clearly in favour of the former and rejects the latter.

I'd be interested to see if you can do this. As far as I can see every time this is looked at seriously the conclusion is resoundingly clear, both approaches (expanding space or galaxies flying apart) are exactly equivalent and every suggestion of an observable difference has at heart some mathematical error. When the co-ordinates are dealt with correctly observables are unchanged, as they should be.

The only contrast between the approaches is a question of which might be the best to guide intuition. When the handle is properly cranked it is clear that they two approaches are just different co-ordinate descriptions of the same physics. If you think you can show otherwise I'd be interested to see your argument, since I can't imagine how this would be possible myself.

This is my promised attempt to show a clear observational difference between the two models. Even if I fail, I hope this attempt will help will be the basis for a clear discussion of the issues at stake.

I will start with the observed redshift (z) that is measured and analyse the famous example of SN1a 1997ff, the supernova observation with z=1.7 that seemed to put the idea of the accelerating expansion rate of the universe on a firm footing.

I will start with some definitions just in case there are some issues with my interpretation of the terminology and semantics used in these discussions and hopefully they will be cleared up in this thread.

Conformal model.
Static spacetime background that does not expand.
Moving objects obey the rules of Special Relativity.
All relative motion is subluminal.

Co-moving model.
Space itself is expanding.
Receding galaxies are not subject to relativistic time dilation as they are at rest with the local space. *
Distant galaxies may be considered to be receding from us at superluminal velocities.

In both models I will assume a low mass density and that space is essentially observed to be flat or very nearly flat. Mass density will be assumed to be homogenous and isotropic on large scales and local concentrations of density such as galaxies will be ignored.

Further assumptions.
An atom of hydrogen here is essentially the same as an atom of hydrogen "there" and the same goes for an atom of hydrogen now and an atom of hydrogen "then".
*The same goes for supernovae as I will be assuming they are ideal standard candles and to keep things ideal supernovae will be assumed to have no local peculiar motion in the coordinate model and remain essentially at rest with the local space.
Initially it will be assumed the rate of expansion is neither accelerating or decelerating and later we will see if that is a reasonable assumption.
Unless otherwise stated assume hypothetical ideal parameters.

On this basis, the observation of the shifted spectrum of z=1.7 will be taken as a pure observational fact. Now if we consider an object that is receding at v/c=1.7 that is not subject ot SR time dilation then there will be an effective time dilation due to non-relativistic Doppler shift due to the distance the object moves away during the interval of the event being observed and this equates to to an observed time dilation of (z+1) =2.7 This is illustrated on the right of the attached diagram. On the left of the diagram is the conformal model. In this model, the observed time of the event is time dilated by a factor of 1.53519 due to Special Relativistic time dilation and by a further factor due to classic Doppler shift to give a total that is also (z+1)=2.7 which is in fact the relativistic Doppler shift. At this point the two models seem to agree with observation. This quote shows that the time dilation of supernovae events corresponds to (z+1).
--------------------------------------------------------------------
http://www.eurekalert.org/features/doe/2001-04/dbnl-tof053102.php
"Twenty-five days later may seem like a long time, but highly redshifted objects are moving away from us so fast that time dilation is large," Nugent remarks. "At a redshift of 1.7, three and a half weeks in our frame of reference is only about nine days of elapsed time for the supernova itself."
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It can be quickly checked that 9*2.7 = 24.3 = 9*(z+1) is in pretty good agreement with Nugent's statement.

This is also true on a more general basis that all supernovae at any redshift (z) basically show this (z+1) time dilation correspondence as shown by this papaers in this FAQ.
--------------------------------------------------------------------
http://www.astro.ucla.edu/~wright/cosmology_faq.html#TD
"This time dilation is a consequence of the standard interpretation of the redshift: a supernova that takes 20 days to decay will appear to take 40 days to decay when observed at redshift z=1. The time dilation has been observed, with 5 different published measurements of this effect in supernova light curves"
--------------------------------------------------------------------

Now this is where the correspondence of the conformal and co-moving models breaks down. If you look at the diagram, there are two curved green lines showing the paths of photons in co-moving space. It can be seen in the diagram that the arrival times no longer correspond to a time dilation of (z+1) and that the start and end of the event is no longer 2.7 but 5.49596 longer than the proper time of the event. A supernova event lasting 9 days in its own frame would be seen as lasting just over 49 days from the Earth rather than the 25 days that was actually observed. This rules out the co-moving model as a viable model as it does not agree with actual observations.

When the co-moving model is ruled out, there are no superluminal recession velocities, and the accelerated expansion of the universe appears to be an artefact of assuming the co-moving model.

 

[Wallace]

Conformal model.
Static spacetime background that does not expand.
Moving objects obey the rules of Special Relativity.
All relative motion is subluminal.

Co-moving model.
Space itself is expanding.
Receding galaxies are not subject to relativistic time dilation as they are at rest with the local space. *
Distant galaxies may be considered to be receding from us at superluminal velocities.

In both models I will assume a low mass density and that space is essentially observed to be flat or very nearly flat. Mass density will be assumed to be homogenous and isotropic on large scales and local concentrations of density such as galaxies will be ignored.

What do you mean by 'low mass density' and that space is flat or near flat? If by the first you mean that
$$\Omega_{total energy} << 1$$

then clearly you cannot have the second. If in fact you have flat or near flat space, i.e.

$$\Omega_{total energy} = 1$$

then you cannot ignore the effects of gravity. In this case the first model fails since there is not gravity in SR (unless you perhaps put some Newtonian gravity in by hand in which case it will work for low redshifts until the Newtonian approximation breaks down). In the second model the effect of gravity is accounted for by altering the rate of the expansion of space.

If you correctly consider both motion and gravity both models will give you the same result if they are correctly formulated. Essentially you can put it like this, in a 'kinematic' interpretation we would define

$$z_{total} = z_{Doppler} + z_{gravity}$$

Note that you have not considered the gravitational effects in your kinematic interpretation sums. Note that the gravitational redshift stretches SN light curves in the same was as Doppler, so you can't observationally distinguish them, redshift is redshift, all that matters in the end is the total.

For a 'conformal co-moving' interpretation we would write

$$z_{total} = z_{change in metric (expansion of space)}$$

however if we correctly run the sums we get the same total redshift in both case, since it is simply a matter of changing co-ordinates. Since the total redshift is all we can measure both interpretations are equivalent mathematically and physically. The only difference between them is the mental picture we have to think of them.

However, the main message is that you cannot ignore gravity no matter what co-ordinate you want to use.

 

[yuiop]

What do you mean by 'low mass density' and that space is flat or near flat?

This a quote from the Ned Wright's cosmology tutorial/FAQ
-----------------------------------------------
"The space-time diagram http://www.astro.ucla.edu/~wright/omega0.gif shows a "zero" (really very low) density cosmological model plotted using the D(now) and t of the Hubble law." http://www.astro.ucla.edu/~wright/cosmo_02.htm
-----------------------------------------------
Ned Wright's statement is pretty close to my statement. My intended meaning is model in which the influence of gravity is almost insignificant. When trying to deduce patterns it is best to start with a simple model and refine it from there. For example when the Gas laws were deduced the effect of intermolecular forces, volume occupied by individual molecules, molecular spin etc were ignored to get to the crux of the matter in defining the ideal gas laws. Starting with a cosmological model based on special relativity in an infinite universe with no acceleration or deceleration it is easy to see that the model very closely actual empirical observations. In fact observable universe shows almost no curvature. So much so it is still not certain whether the universe is closed, flat or open. The Omega(total) is very close to 1.0 and the current best estimate is 1.01 which is a closed universe, but only marginally so and almost indistinguishable from a flat infinite universe. I mentioned in old thread that in an infinite universe gravitational collapse or deceleration is not possible because there is no preferred direction for any given galaxy to gravitate towards, when local density fluctutions are ignored. In other words, in an infinite universe, there is no large scale gravitation, only local clumping. I personally believe that the universe is not infinite but is significantly larger than our visible universe and very closely aproximates the infinite case which is largely in agreement with the observed best estimate of a value for Omega(total) of 1.01

In the conformal (Special Relativistic) model, receding galaxies basically move as projectiles under there own momentum as opposed to being carried along, as if embedded in expanding space as in the comoving model. Starting with an assumption of negligable gravitational influence in the Special Relativistic model the predictions are remarkably in accord with what is actually observed. The time dilation observed in supernovae explosions is almost exactly in agreement with the Special Relativistic conformal model without any additional modifications due to gravity or acceleration or deceleration. The Special Relativistic model without a cosmological constant or dark energy does not have the disadvantage of having to account for the remarkable coincidence that is known as "the flatness problem".
------------------------------------------------------------------------------------
Adding only 1 gm/cc to this 447 sextillion gm/cc causes the Big Crunch to be right now! Taking away 1 gm/cc gives a model with Ω that is too low for our observations. Thus the density 1 ns after the Big Bang was set to an accuracy of better than 1 part in 447 sextillion. Even earlier it was set to an accuracy better than 1 part in 1059!
http://www.astro.ucla.edu/~wright/cosmo_03.htm
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The conformal model, without dark energy only requires that the total universe is significantly larger than the visible universe for the universe to appear flat and the flatness of the universe is nothing special about the epoch we happen to be in. The flatness in the conformal model comes about as a result of rapid inflation in the very early universe and has remained essentially flat since then until now.

The comoving model on the other hand can not duplicate the time dilation observations no matter what mix of matter and dark energy is postulated. As I showed in my previous post the effective time dilation of a pair of events in comoving space depends on classic doppler shift due to the recession of the supernova and an additional factor due to the expansion of space "stretching" the time interval in the same way that the wavelength of a photon is stretched. Since the classic doppler shift due to recession can completely account for the time dilation by itself in the comoving model, any non zero effect due to the expansion of space makes things worse. The time dilation due to the expansion of space can also account completely for the observed time dilation by itself in the comoving model, but the recession velocity makes things worse.

The comoving model requires:
dark energy,
space as expanding medium that is basically the old aether in disguise,
a coincidence of cosmological proportions (the flatness problem)

The Special Relativistic model can account for all the observations, without requiring any of the fanciful assumptions of the comoving model.

When gravity is taken into account for the Special Relativity model the difference is almost negligable if our visible universe is a small part of a larger (but not necessarily infinite) universe. However, the slight excess of having a value of Omega(total)=1.01 rather than a perfectly flat value of 1.00 means that the universe will eventually collapse in the absence of dark energy.

Whatever parameters are used for mass and dark energy in the comoving model, the observed time dilations can not be reproduced. That is why the comoving model fails. Not just because it requires a lot of fanciful assumptions but because it simply does not match observations. The counterproof would be to demonstrate that the comoving model can match the observed time dilations. Can you do that?

 

[Wallace]

In fact observable universe shows almost no curvature.

The observable Universe shows almost no spatial curvature when describe in co-moving co-ordinates. Spatial curvature is a co-ordinate dependent quantity and cannot be independently defined without reference to co-ordinates. The total curvature of the Universe is most definitely observed to be non-zero.

So much so it is still not certain whether the universe is closed, flat or open. The Omega(total) is very close to 1.0 and the current best estimate is 1.01 which is a closed universe, but only marginally so and almost indistinguishable from a flat infinite universe.

Given the way Omega is defined, which requires the use of co-moving co-ordinates in the FRW model, an omega close to unity indicates we cannot ignore gravity since the Universe sufficiently dense. In these co-ordinates, if the universe was sufficiently underdense such that we can ignore gravity then Omega << 1. We do not measure Omega! We measure others things and put those observables into a model. The model requires we define a co-ordinate system and in that system we infer a value of Omega.

I mentioned in old thread that in an infinite universe gravitational collapse or deceleration is not possible because there is no preferred direction for any given galaxy to gravitate towards, when local density fluctutions are ignored. In other words, in an infinite universe, there is no large scale gravitation, only local clumping.

I know there have been some poor threads on this topic around here. This statement above is simply incorrect, an infinite universe can of course decelerate. Decceleration is the decrease in relative velocity between all points in the Universe. It does not mean that all points move towards the centre. You can define any arbitrary co-ordinate centre and declare that the whole Universe is accelerating towards that centre. If someone defined this centre to be elsewhere the result is that the relative motion between any pair of particles you choose is the same regardless of where you declare the co-ordinate centre to be. There is nothing about an infinite Universe that precludes deceleration in either Newtonian or Einsteinian gravity.

In the conformal (Special Relativistic) model, receding galaxies basically move as projectiles under there own momentum as opposed to being carried along, as if embedded in expanding space as in the comoving model. Starting with an assumption of negligable gravitational influence in the Special Relativistic model the predictions are remarkably in accord with what is actually observed.

This 'coasting' or 'Milne' model is in fact a poor fit to the data, ruled out by several sigma. The data do agree with this model at low redshift, but that is because all models look like this model at low redshift. At high redshift the Supernovae data diverges from this curve. The fit is far worse for structure formation. In coasting models there is simply not enough time for the amplitude of structure observed to form. So no, this model is not in accordance with the data.

The time dilation observed in supernovae explosions is almost exactly in agreement with the Special Relativistic conformal model without any additional modifications due to gravity or acceleration or deceleration.

All causes of redshift are causes of time dilation, so all model with have exact agreement between time dilation and redshift. However, what matter for the SN data is how the distance modulus varies with redshift. For the SR model the prediction does not match the data.

The Special Relativistic model without a cosmological constant or dark energy does not have the disadvantage of having to account for the remarkable coincidence that is known as "the flatness problem".

Your right, there is no flatness problem in this model because the Universe is non flat in this model (in this context 'flat' means spatial flatness in FRW co-ordinates). Pity then that the data points to flatness, ruling this model out.

The comoving model on the other hand can not duplicate the time dilation observations no matter what mix of matter and dark energy is postulated. As I showed in my previous post the effective time dilation of a pair of events in comoving space depends on classic doppler shift due to the recession of the supernova and an additional factor due to the expansion of space "stretching" the time interval in the same way that the wavelength of a photon is stretched. Since the classic doppler shift due to recession can completely account for the time dilation by itself in the comoving model, any non zero effect due to the expansion of space makes things worse. The time dilation due to the expansion of space can also account completely for the observed time dilation by itself in the comoving model, but the recession velocity makes things worse.

You have to focus on what we measure. We measure a redshift. We can't independently know the speed to compare to a doppler shift formula. As I say, time dilation and redshift are always hand in hand regardless of whether the redshift is 'caused' by doppler or gravitational effects. Time dilation and redshift are one and the same thing.

What we measure are properties of the Universe as a function of redshift. We have to predict these properties from the model then compare to observations. We can easily make these predictions for a Universe with vanishing density, and it is a poor fit. A model with matter and dark energy is a good, that is the reason it is the currently favored model!

The comoving model requires:
dark energy,
space as expanding medium that is basically the old aether in disguise,
a coincidence of cosmological proportions (the flatness problem)

There are many threads linking to many papers that explain that 'expanding space' is a co-ordinate dependent thing. The same physics (and same observable implications) can be described in equivalent co-ordinate systems in which there is no expanding space. It is a mistake to think that the standard cosmological model requires an aether like medium. This is simply not true.

The Special Relativistic model can account for all the observations, without requiring any of the fanciful assumptions of the comoving model.

No it can't. If it fitted the data it would be the concordance model.

Whatever parameters are used for mass and dark energy in the comoving model, the observed time dilations can not be reproduced. That is why the comoving model fails. Not just because it requires a lot of fanciful assumptions but because it simply does not match observations. The counterproof would be to demonstrate that the comoving model can match the observed time dilations. Can you do that?

Yes, all processes that cause redshift cause time dilation. They are equivalent things, the correspondence is always exact in every model. As I say, cosmological observations observe the nature of the Universe as a function of redshift, that is how models may be distinguished.

 

[chronon]

If 'special relativistic' is taken to mean the (0,0) model of the universe then Wallace is right. It is 2 sigma away from the best fit of the supernova data, and is not flat in comoving coordinates as is required by the CMBR data.

However, if you take a more general meaning of 'special relativistic' to mean choosing a coordinate system within in which the speed of light is the limiting speed then Wallace's arguments don't apply.

However, the statement that redshift must always agree with observed time dilation still stands, as redshift is simply the observed time dilation between wavefronts. Time dilation=z+1. So kev's argument would also imply that the redshift of the galaxy should be greater.

What is wrong with kev's argument is that it is double counting. One way of getting the redshift is to use the non-relativistic formula for the galaxy moving at superluminal speeds. But the usual way of arguing is that the galaxies are stationary in space, and so don't have any redshift due to motion, but that the space the light is traveling through expands, and that is what causes the redshift.

 

[Wallace]

However, if you take a more general meaning of 'special relativistic' to mean choosing a coordinate system within in which the speed of light is the limiting speed then Wallace's arguments don't apply.

Unfortunately this kind of discussion always encounters the problem that we need to be very precise in the meaning of the terms we use. If we have slightly different meanings for terms it is easy to think we disagree when at base we do not.

In this case there are so problems with your statement. In defining speed we need to define the rate of change of some distance with respect to some time. Neither distance or time are co-ordinate independent and at cosmological distances it takes a (long!) finite time to measure a distance!

Special and General Relativity both specify that motion through an inertial frame must be sub-luminal. This is because due the equivalence principle allows us to specify a Minkowski tangent frame to any inertial frame in this we can unambiguously measure speed and know that it must be sub luminal.

We cannot make this measurement therefore at cosmological distances, and hence any speed we define depends entirely on the co-ordinate system we choose. We can construct a conformal co-ordinate system (see recent papers by Chodorowski and Lewis, Francis, James, Kwan, Barnes) in which recession velocities are sub-luminal, or we can use the FRW co-ordinates in which they are superluminal. The physics and the observables are all the same, the co-ordinates are arbitrary.

However, a Universe can only be considered to 'Special Relativistic' if the effects of gravity can be ignored. A metric that is conformally related to the metric of SR is not SR. Again, see the recent papers I mentioned. The only case in which the conformally Minkowski metric becomes equivalent to SR is when the energy density goes to zero, in which case GR and SR become equivalent.

Could you be clearer therefore about what you mean by the above statement, and how it makes my previous arguments invalid? As I say, on these matter unfortunately pedantry is necessary to avoid having false disagreements.

However, the statement that redshift must always agree with observed time dilation still stands, as redshift is simply the observed time dilation between wavefronts. Time dilation=z+1. So kev's argument would also imply that the redshift of the galaxy should be greater.

Agreed.

What is wrong with kev's argument is that it is double counting. One way of getting the redshift is to use the non-relativistic formula for the galaxy moving at superluminal speeds.

I'm not sure that this works? I'd be interested to see if you could demonstrate this mathematically, but what I seen in papers (and played around with myself) is that you can show that the 'superluminalness' of the recession speed can be accounted for due to the effects of gravity. See for instance the recent paper on rocket ranging by Lewis et al. It's the gravity that is important in combination with motion rather than trying to hack some unphysical numbers into a formula to try and account for the redshift by motion alone.

But the usual way of arguing is that the galaxies are stationary in space, and so don't have any redshift due to motion, but that the space the light is traveling through expands, and that is what causes the redshift.

Yep, and the amount of 'expansion of space' a phenomenon purely co-ordinate dependent, is dictated by the energy content of the Universe, indicating once again that it is gravity that is the key.

 

[chronon]

Unfortunately this kind of discussion always encounters the problem that we need to be very precise in the meaning of the terms we use. If we have slightly different meanings for terms it is easy to think we disagree when at base we do not.

You're right of course, I shouldn't postulate new coordinate systems without defining exactly what they are. Indeed there are too many coordinate systems already. I think that radar coordinates might do the job, but I'm really thinking in terms of a very long ruler. I acknowledge that I then need to show that such a ruler agrees with the General relativity and to work out what the coordinate system actually gives. Feel free to ignore ruler coordinates in what follows.

There are then five possible coordinate systems.

Fully Conformal
Partially Conformal (as in Lewis et. al. http://arxiv.org/PS_cache/arxiv/pdf/0805/0805.2197v1.pdf)
Radar
Ruler
Comoving

Assume lambda=0 throughout. In the omega=0 case the Fully Conformal, Ruler and Radar systems will agree. In the omega>0 case the five systems may well be all different. Fully conformal and Radar will give subluminal velocities throughout, whilst Partially Conformal an Comoving allow for superluminal velocities.

I'm not sure that this works? I'd be interested to see if you could demonstrate this mathematically, but what I seen in papers (and played around with myself) is that you can show that the 'superluminalness' of the recession speed can be accounted for due to the effects of gravity. See for instance the recent paper on rocket ranging by Lewis et al. It's the gravity that is important in combination with motion rather than trying to hack some unphysical numbers into a formula to try and account for the redshift by motion alone.

kev's arguments actually apply to the (0,0) universe, that is the one illustrated in the diagrams at http://www.astro.ucla.edu/~wright/cosmo_02.htm#DH, which show comoving and conformal systems. The system allowing superluminal velocities is comoving rather than Partially Conformal. For omega>0 the diagram http://www.astro.ucla.edu/~wright/cosmo_03.htm#MSTD looks very similar, so the superluminal velocities in this coordinate system can't be put down to the effects of gravity.

Then there's the question of which coordinate system this is being compared with - the one which forbids superluminal velocities. kev talks of the (Fully) conformal system, but this is a pretty weird system, in that it does not show the deceleration which takes place due to gravity. I would think that radar coordinates might be better.

 

[jonmtkisco]

Here's an off-the-wall idea about an effect that might explain the appearance of superluminal recession even if (hypothetically) the proper distance between the observer and the galaxy is not actually increasing superliminally.

There are two contributors to redshift: The Doppler Effect, and gravitational redshift. Gravitational redshift occurs when light has to climb its way out of a gravity well.

If we assume the cosmic fluid (matter) is homogeneous, then it does not seem to contain any such gravity well. However, if we look far back in time, the cosmic fluid was much denser than today. In that sense, any light emitted long ago began in a universe-sized gravity well (relative to current density), and has been climbing out of that well ever since, as background density increases.

So my question is whether it is possible for a timelike gravity gradient to cause a photon to experience progressive gravitational redshift over time and distance. Since light travels on null geodesics, perhaps it is equally susceptible to both timelike and spacelike gravitational gradients.

On the other hand, the photon is traveling through space as well as time, so perhaps it is sufficient to explain the effect in spacelike terms alone: each meter of space the photon passes through is slightly less dense than the previous meter (on average), so to the photon experiences simply a normal spacelike gradient. The latter explanation seems more straightforward.

Just a thought.

Jon

 

[jonmtkisco]

Oops, obviously I meant this sentence to read as corrected here:

In that sense, any light emitted long ago began in a universe-sized gravity well (relative to current density), and has been climbing out of that well ever since, as background density decreases.

And I meant "superluminally", not "superliminally".

Jon

 

[Wallace]

Here's an off-the-wall idea about an effect that might explain the appearance of superluminal recession even if (hypothetically) the proper distance between the observer and the galaxy is not actually increasing superliminally.

There are two contributors to redshift: The Doppler Effect, and gravitational redshift. Gravitational redshift occurs when light has to climb its way out of a gravity well.

If we assume the cosmic fluid (matter) is homogeneous, then it does not seem to contain any such gravity well. However, if we look far back in time, the cosmic fluid was much denser than today. In that sense, any light emitted long ago began in a universe-sized gravity well (relative to current density), and has been climbing out of that well ever since, as background density increases.

So my question is whether it is possible for a timelike gravity gradient to cause a photon to experience progressive gravitational redshift over time and distance. Since light travels on null geodesics, perhaps it is equally susceptible to both timelike and spacelike gravitational gradients.

On the other hand, the photon is traveling through space as well as time, so perhaps it is sufficient to explain the effect in spacelike terms alone: each meter of space the photon passes through is slightly less dense than the previous meter (on average), so to the photon experiences simply a normal spacelike gradient. The latter explanation seems more straightforward.

Just a thought.

Jon

I don't think this is 'off the wall'. If you have a look at the fundamental derivation of redshift in an FRW universe in any standard text (I'm looking at Hartle at the moment but any should do, Peacock, Harrison, Peebles...) then this is pretty much what you find, i.e. the only non zero gradient of the photon energy as a function of some affine parameter is the time derivative, due to homogeneity in the spatial dimensions at any constant time slice. Of course, as has been discussed ad norsium, this doesn't fundamentally mean anything, it's just how it works in these co-ordinates.

 

[jonmtkisco]

Hi Wallace,
I bought Hartle's textbook some months ago on your recommendation. I'm looking at Section 18.2 The Cosmological Redshift. As I read it, it seems to describe cosmological redshift only in terms of the size increase of the scale factor. I don't see any specific mention of an additional effect caused by the gravity gradient of the cosmic fluid's decreasing density over time. For example:

"In an expanding universe where a(t) grows with t, the ratio a(t_e) / a(t₀) will be less than 1 and the received frequency w₀ less than the emitted one w_e. That is the cosmological redshift. As the universe expands, the frequency of the photon decreases, and its wavelength increases linearly with the scale factor a(t)."

Section 9.2 describes gravitational redshift in terms of the Schwarzschild metric, but not in terms of the FLRW metric.

Can you please point me to Hartle's description of how the temporal gravity gradient of an FLRW cosmic fluid affects the cosmological redshift?

Jon

 

[Wallace]

Okay, looking back I somewhat misread your comment. I was merely stating that it is clearly the change in time of the metric that causes redshfit in the FLRW line element. Since the metric encodes the effects of gravity (the metric IS gravity if it is a valid solution to the field equations) then in effect the metric already encodes the physical effects you mention. I don't think it is as direct as you state, at least not a term in the equation you can point to, but the effect is in there in the form of a(t).

 

[yuiop]

Hi all,

I have given the subject some further thought and research and concede the point by Wallace and others that causes of redshift are also causes of time dilation which ever model you choose. The various models only differ in the assumed recession velocity of the galaxies or supernovae at the time of emmision and in distances at the time of emmision calculated from flux and luminosity measurements.

The red tear drop curves in the attached diagram are the light paths in co-moving model with gravity assuming (t/to)^(2/3 )with Ωo = 1. They are plotted using the equation (derived by George in the Relativity forum) of x = n(t^(1-1/n)-t) with a value of 3 for n giving x=3(t^(2/3)-t). This is an exact fit for the curve given by Ned Wright here http://www.astro.ucla.edu/~wright/cosmo200.gif

The blue tear drop curves in the attached diagram are the light paths in a low density model with negligable density which is basically the the match for the curve given by Ned Wright here http://www.astro.ucla.edu/~wright/omega0.gif and is plotted using a value of n=1000 in the x = n(t^(1-1/n)-t) equation. There is not much difference between the curves for n=1000 and n approaching infinity.

The interesting part is that both models satisfy the requirement that (z+1) = (to/te) and both satisfy the (z+1) = time dilation factor. In the diagram for z=1.7 the time of emmission te is represented by tA and is numericaly equal to 0.3707 and when multiplied by (z+1) =2.7 this gives a value of 1 which is where t0 is situated on the diagram. At time tA the velocity of the supernovae at the time of emmision in the low density model is about 0.99c while in the mass dominated model (t/to)^(2/3) the velocity at the time of emmision is 0.85c. Note that in neither model is the velocity equal to z.

The really interesting part of the diagram is that if you look at the middle blue and red teardrop curves terminating at epoch t1, the curves cross over at about z=0.5. The distances at the time of emmision in the low density model are further away (and darker) below z=0.5 and nearer (and brighter) at redshifts above z=0.5 than would be expected for the matter dominated model. This is basically the observation that is at the root of the conclusion that the rate of expansion is accelerating. The low density model seems to produce the same brightness anomally relative to the assumed (t/to)^(2/3) model without requiring the rate of expansion to be accelerating.

To fill in some more of the details of the diagram, the coloured curved lines going out to the right (labelled vs) are the trajectories of galaxies in the expanding models. The straight black lines on the right is the geometry of a model without gravity, relativity or expanding space where light travels in straight lines and is not representative of anything physical but gives something to compare the physical models with. The lines in the green section on the left is the equivalent special relativity model of the low density model at z=1.7

 

[Wallace]

Your on the right track, except that an empty Universe does not look the same as an accelerating one. Everytime the Supernovae data is analysed this is examined and it simply doesn't fit the data points. It might look similar to an accelerating model, but not similar enough to fit the data nearly so well.

 

[jonmtkisco]

Hi Wallace,

Since the metric encodes the effects of gravity (the metric IS gravity if it is a valid solution to the field equations) then in effect the metric already encodes the physical effects you mention. I don't think it is as direct as you state, at least not a term in the equation you can point to, but the effect is in there in the form of a(t).

To be fair, the FLRW metric for a flat matter-only universe encodes only that it is physically stretching at the escape velocity of its contents. So in that very basic sense, gravity is encoded in a(t).

But the off-the-wall idea here is that there is an additional component of gravitational redshift beyond that attributable directly to the stretching of the scale factor. A factor which reflects the experience of a photon moving through regions that are not only stretched but also are characterized by a progressively diminishing gravity well in the cosmic fluid.

I don't see any reference to that in Hartle or Peebles, or in pages 367-68 of Hobson, Efstathiou & Lasenby (2005).

Jon

 

[Wallace]

So you are proposing that there is an additional effect due to gravity that is not described by General Relativity? In that case then I agree, it is an off the wall idea and not surprisingly doesn't appear in Hobson, Peebles etc.

 

[jonmtkisco]

Hi Wallace,
The hypothetical effect I'm asking about is based on GR and as far as I can tell would be completely consistent with it. It's nothing more than identifying a (possibly overlooked?) situation where plain old gravitational redshift could occur, and applying standard GR and the FLRW metric to calculate the answer.

I don't see how it's any more radical than, say, the way the Lewis & Francis paper on Radar Ranging innovatively applies Gauss' Law to explain the "overshoot" of a returning radar signal. The authors don't attribute that idea to another author.

If you believe that the hypothetical application of GR I'm asking about is fundamentally inconsistent with GR, I'd appreciate if you could explain why.

Jon

 

[Wallace]

Because the metric already encodes everything that GR has to say about gravity! Redshift (and this is talking about the effect due to the homogeneous universe only) is determined by the difference in scale factor at the cosmic time of emission and reception. All effects of gravity the GR know about have already been included in this, you can't add any other effects and still be consistent with GR. This has not been 'overlooked', it is already included, you can't add it in twice.

The paper you mention makes a Newtonian analogy to explain why the effects of GR are maybe not as strange as might first be thought in the situation being examined. By thinking about the qualitative behavior you would get by using Newtonian gravity, understanding the GR result becomes easier. They certainly are not suggesting that you actually use Gauss's law to add additional effects in that have been 'overlooked'. The numbers are all crunched simply from the metric and the geodesic equation, Guass's law is invoked only as a guide to understanding, not used to calculate the numbers since although the qualitative answer would be the same (i,e. whether the rocker over or undershoots or makes a symmetric journey), the specific numbers would be wrong (i.e the Newtonian 'Gauss Law' results differs from the GR) if the journey went to a reasonable cosmological distance. To see an example of this see Barnes et al 'Joining the Hubble Flow' where the GR and Newtonian curves for one tethered galaxy model are shown. The behavior has the same form but the specifics are different.

If you want to think about cosmological redshift being produced in the manner you describe then that's fine, I think it's a reasonable mental picture. My original comment made the point that your description was a reasonable description of what the GR equations tell you. This effect however is already included in those equations and can't be double counted by adding it in again.

 

[jonmtkisco]

Hi Wallace,
Hmmm. OK, thanks for the explanation.

It just seems to me that if you consider an SR recession of two rockets away from each other in an almost-empty FLRW universe, and the rockets use their motors to maintain their recession speed (relative to each other) as a function of time such that it exactly duplicates the (subluminal) relative motion of two comoving particles in an FLRW Omega(m)=1 universe, you'd get the same Doppler redshift over time as you would get between the two particles in the FLRW Omega(m)=1 universe at each respective proper distance. Maybe that's not true.

Jon

 

[Wallace]

I'm pretty sure that's not true, but it would be interesting to see what the results would be. In the end though, to know for sure there's only one way, and that is to do the calculation. I think you're understanding this stuff pretty well, but as I've said before, to get to the next level you really need to start playing with the equations. Analogies and concepts will only get you so far and will lead you astray if you try and push them too far, which I think you are doing.

Try and re-create some of the results in the papers about this stuff, most of them go through from the basics and if you have Hartle that should give you all you need that isn't covered. I'd be interested to know the results from the thought experiment you mention above, but without cranking the handle we're just punching smoke to try and work out the answer just from 'mental pictures' alone. It's always better to know the result from an exact calculation, and then try and understand in simpler terms what the results are telling you. That's pretty much the format of most of the work on this. If you want to push something to an area not explicitly covered already then unfortunately I don't think there are any shortcuts to getting there.