The Mystery of Superluminal Recession Velocities in Cosmology

[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).
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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.
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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"
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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
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"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
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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".
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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.

 

[yuiop]

...
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

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.

I found this interesting question:
------------------------------------------------------------------------------
Optional Problem 10: “Minkowski space in disguise” (hard!): Show by a clever choice of coordinates that the FRW metric with ΩΛ =Ωm =Ωγ =0, Ωk = 1 (this is the special case with a(t)= t, k = −1, corresponding to an empty and maximally open universe) is simply the Minkowski metric in disguise.
------------------------------------------------------------------------------

in this problem set http://ocw.mit.edu/NR/rdonlyres/4973019C-C55D-43D0-81AF-6FDA7D4444C7/0/ps8.pdf that seems to be related to the issue you are discussing. Anyone able to answer that question?

 

[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).

I'd like to get a more specific understanding of what you said here.

Is, or is not, the normal GR cosmological redshift calculation, proportional to a(t), divisible into two distinct, directly additive components: (1) the gravitational redshift proportional to the temporal change in cosmic density from emission event to reception event, and (2) SR Doppler redshift proportional to recession velocity as between the emitter and receiver? This question assumes that the gravitational redshift component is calculated using only the standard form of the GR gravitational redshift equation in the Schwarzschild metric, and the Doppler redshift component is calculated using only the standard SR Doppler redshift equation.

Jon

 

[yuiop]

Hi Wallace,

I'd like to get a more specific understanding of what you said here.

Is, or is not, the normal GR cosmological redshift calculation, proportional to a(t), divisible into two distinct, directly additive components: (1) the gravitational redshift proportional to the temporal change in cosmic density from emission event to reception event, and (2) SR Doppler redshift proportional to recession velocity as between the emitter and receiver? This question assumes that the gravitational redshift component is calculated using only the standard form of the GR gravitational redshift equation in the Schwarzschild metric, and the Doppler redshift component is calculated using only the standard SR Doppler redshift equation.

Jon

Hi Jon,
Hope you don't mind me jumping in and adding my 2 cents worth. As I understand it the redshift is due to the "expansion of space" stretching the wavelength of the photon during its journey and a component that is due to the motion of the emitter away from the receiver during the interval that the photon is emitted which is equivalent to classic doppler red shift without the SR time dilation component. There is no SR time dilation because the source is stationary with respect to the local space. However, you can swap coordinate systems and treat the space as not expanding so there is no stretching of the wavelength during its travels and then SR time dilation does apply because in this coordinate system (conformal) the source is moving relative to the local space (which remains static) and the doppler shift is calculated as per SR doppler shift. The equivalence of the SR conformal system with the co-movingsystem is illustrated by Ned Wright here http://www.astro.ucla.edu/~wright/cosmo_02.htm The end result is the same, but both of the systems Ned Wright is using do not have the the effect of gravity included, which has to factored in in a non empty universe.

 

[Wallace]

Hi Wallace,

I'd like to get a more specific understanding of what you said here.

Is, or is not, the normal GR cosmological redshift calculation, proportional to a(t), divisible into two distinct, directly additive components: (1) the gravitational redshift proportional to the temporal change in cosmic density from emission event to reception event, and (2) SR Doppler redshift proportional to recession velocity as between the emitter and receiver? This question assumes that the gravitational redshift component is calculated using only the standard form of the GR gravitational redshift equation in the Schwarzschild metric, and the Doppler redshift component is calculated using only the standard SR Doppler redshift equation.

Jon

No, as Kev referred to, if you use the FRW co-ordinates then a(t) gives you the redshift. That's it. You can slice up how a(t) came to be what it is if you like, but the effects you refer to are already in there. In these co-ordinates co-moving particles are at rest, so there is no Doppler shift calculation. As we know, this physical effect of Doppler shift has already been included by the choice of co-ordinates.

This is one of the many reasons why FRW co-ordinate are so useful. You just have to remember not to take phrases like 'the expansion of space' literally!

By the way, I'm not sure that you recover the same redshift as FRW if you use an SR doppler shift with an added gravitational redshift using a Schwarzschild metric? Maybe you do, I haven't seen this demonstrated so neatly though. I know there have been arguments that in principle redshift can be divided up this way but again, while this is a reasonable qualitative argument, I don't know that you get exactly the same answer. GR is non linear so you get to trouble trying to linearly add things if you're not careful. Have you seen anyone show this relationship explicitly? I'd be interested to see if this had been looked at in detail.

 

[jonmtkisco]

Hi Wallace,
Yes as I went back and reread some papers I realized that I must have originally picked up the idea from Alan Whiting's 4/04 http://arxiv.org/abs/astro-ph/0404095" . He takes a couple of different approaches to dividing the cosmic redshift into its component parts.

He applies a sort of Newtonian Shell Theorem approach to calculating the gravitational redshift component, and combining it with SR Doppler redshift he recovers the cosmic redshift formula in a backhanded way but says it is a special case for a flat FRW universe with Lambda=0 only.
His Shell Theorem approach seems to measure the difference between the total matter density now and zero matter density now.

As far as I can tell he does NOT approach it specifically from the perspective of the temporal/spatial density gradient from Event E to Event O, which it seems to me is more likely to be fruitful.

Jon

 

[Wallace]

Whitings formula is, as I say, only approximate. It gives you back a similar behavior but it quantitatively different from the GR approach. As I said earlier, this was shown in Barnes et al "Joining the Hubble Flow". It works on the level of conceptual understanding.

You asked a specific question, whether SR doppler shift + a gravitational redshift calculated with the Schwarzschild metric gives you the same result as FRW. I don't think this is the case, but Whitings Newtonian result will certainly differ from both FRW and this approach you describe.

I'm not saying Whiting was 'wrong', his paper was intended to demonstrate a point of understanding. He probably acknowledges somewhere in there that his formulas would surely break down at some point. You need to separate quantitative calculations from lines of argument intended to give a greater intuiative qualitative understanding.

 

[jonmtkisco]

He Wallace,

... Whitings Newtonian result will certainly differ from both FRW and this approach you describe.

Well I already suggested that Whiting's result doesn't seem fruitful... He compares total density NOW to zero density NOW. That has nothing directly to do with the past worldline of a photon emitted long ago. I made it clear that I was not saying his result was "correct." Why would it be?

What I suggested is that my approach to attacking the problem makes more sense than his. Therefore by definition it yields a different result.

Previously you scolded me for not combining quantitative calculations with intuitive lines of argument; now you scold me (without basis) for not keeping them separate.

I think MY idea might be original and it definitely is interesting enough to warrant further analysis.

Jon

 

[Wallace]

He Wallace,

Well I already suggested that Whiting's result doesn't seem fruitful... He compares total density NOW to zero density NOW. That has nothing directly to do with the past worldline of a photon emitted long ago. I made it clear that I was not saying his result was "correct." Why would it be?

Whitings 'results' are perfectly fruitful in terms of achieving what he sets out to do. He his not trying to re-derive cosmological redshift.

What I suggested is that my approach to attacking the problem makes more sense than his. Therefore by definition it yields a different result.

What is the problem?? We know perfectly well how to calculate cosmological redshift. Whiting is not trying to re-solve this problem. I'm not sure why you would even compare your suggestion to Whitings paper, which is clearly purely pedagogical.

Previously you scolded me for not combining quantitative calculations with intuitive lines of argument; now you scold me (without basis) for not keeping them separate.

What I'm stressing is that while they both need to work together to best approach a problem, in the end you need to know the difference. If you want to compare to data you need to use a proper calculation. If you want to talk about how best to teach cosmology to students you need to go beyond just saying "work out the maths". This is why you see the kind of co-ordinate transformations etc that are use in a lot of the literature on this stuff. The FRW metric is always the easiest to use to get an numerical answer, but can be a little tricky and confusing conceptually. Con-formal co-ordiantes or Newtonian analogies etc can be a useful way of re-arranging the maths in order that the equations take a more intuitive form, i.e. 'this term represents this, that term represents that'. Of course, no one (included Chodorowksi etc) is actually suggesting that these representations should actually be used by professional cosmologist when assessing data. Clearly the FRW metric is most suitable for that. It is a conceptual exercise.

I think MY idea might be original and it definitely is interesting enough to warrant further analysis.

Jon

YOUR idea is, as I said originally, a rough qualitative description of how redshift is calculated. I'm not quite sure what is particularly original about it? Indeed, what 'further analysis' do you think can be done on this idea? If you use it to motivate a different mathematical method for calculating redshift then you already know the answer that you must end up with. Possibly you can re-calculate the answer through a set of equations that doesn't come straight from the FRW metric but instead is the mathematically description of your words. If this set of equations can be shown to be equivalent to FRW then great, you've come up with a new co-ordinate transformation which might well be an interesting way of thinking about the issue. If you get a different result then the equations might be an approximation, or possibly simply wrong. Certainly it sounds as though you could describe your proposal mathematically such that you do get the right answer.

No one is going to do this analysis for you however, and if you really want to claim you have surpassed Whiting paper then you going to have to demonstrate that with something a lot more concrete than what you have. Although as I say, I think you are wildly misinterpreting the aims of conclusions of that paper.

Lets get this straight. Are you suggesting a new way of calculating redshift such that previous methods are wrong and yours will be correct? Are you instead suggesting a new way of thinking about how redshift occurs which would lead to a new form of equations that give you the correct answer? Are you suggesting a new qualitative description that isn't intended to get a correct numerical answer? None of the above? I think if you can make this clearer it would greatly aid the discussion.

 

[jonmtkisco]

Hi Wallace,

Whitings 'results' are perfectly fruitful in terms of achieving what he sets out to do. He his not trying to re-derive cosmological redshift.

Scold, scold, scold.

First you scold me for supposedly swallowing (not!) that Whiting's result is a correct derivation of cosmic redshift. Now you scold me because, although he did not succeed, you don't think I recognize that he never really tried. In lieu of more scolding, can't we just agree that my proposal suggests a way to attempt this and his proposal doesn't?

Possibly you can re-calculate the answer through a set of equations that doesn't come straight from the FRW metric but instead is the mathematically description of your words. If this set of equations can be shown to be equivalent to FRW then great, you've come up with a new co-ordinate transformation. If you get a different result then the equations might be an approximation, or possibly simply wrong. Certainly it sounds as though you could describe your proposal mathematically such that you do get the right answer.

Hurray! I might be right! Well if you don't want to be the first to find out, that's your prerogative. If you're interested in the subject matter, you might enjoy doing the calculations. It's obvious you are better at it than I am. If you're not interested, I'll struggle with it myself. If there's any merit to the idea, eventually someone will include it in a published paper. If the idea sucks, then it will end up on the scrap heap along with a lot of good company.

Jon

 

[Wallace]

Jon, I am not 'scolding' you. If you don't want feedback then why post your ideas in the first place? I love how you trawl through my whole post ignoring the meat of it to highlight one sentence where you feel that you have had some kind of victory. As if this was some kind of online debating competition. I don't get it. You will make much more progress much more quickly if you are willing to take on board advice and criticism.

 

[jonmtkisco]

Hi Wallace,
So the meat of your post was that my idea is wrong? Or was the meat of it that no idea is really worth discussing if it it isn't accompanied by a complete and accurate set of equations?

I have no interest in scoring debating points. My interest is solely in gaining a deeper understanding of the subject matter. I have greatly appreciated your substantive answers, but often they have been diminished by excessive scolding.

Alan Whiting over 4 years ago suggested at least the possibility that cosmic redshift might be cleanly separable into gravitational and Doppler components. I for one am interested in pursuing the substance of that inquiry. If you share that interest, you haven't made it very clear.

Jon

 

[Wallace]

Hi Wallace,
So the meat of your post was that my idea is wrong? Or was the meat of it that no idea is really worth discussing if it it isn't accompanied by a complete and accurate set of equations?

If you want to discuss whether you can correctly split redshifts into different components then of course this needs to be accompanied by the equations describing this split. I'm not sure why that can be construed as being a controversial statement? It either is or isn't possible. From memory I think this was actually done by Gron and Elgaroy in a paper a few years ago. Can't remember the exact reference but I think it's in there somewhere. I could be mis-remembering though.

Alan Whiting over 4 years ago suggested at least the possibility that cosmic redshift might be cleanly separable into gravitational and Doppler components. I for one am interested in pursuing the substance of that inquiry. If you share that interest, you haven't made it very clear.

Again, I think you are misinterpreting Whitings paper. He was not suggesting that the equations he derived correctly split the redshift in this way. It was only conceptual, to demonstrate the illusionary nature of 'expansion of space'. This redshift split was not really the essence of Whitings paper.

I am always interested in sensibly formulated discussions of cosmology, I'm just trying to unpick what exactly you are aiming at here, since it isn't clear. I know it's clear to you, because you wrote it, but surely asking for clarification when it isn't clear is not 'scolding'. If something isn't clear it is never entirely the fault of the author or the reader!

So anyway it sounds like you are trying to formulate a way to correctly calculate cosmological redshift with an equation that neatly splits into a motion and a gravitational part. This is an interesting question, but Whiting is not the place to start, since his Newtonian approach is guaranteed not to do this (since that is not what is was intended to do). You'd need to start from GR somewhere. As I say I think there is a Gron and Elgaroy paper that does this, maybe check that out (they have a few on this topic I think so it might take some searching).

Edit:

I found it! Turns out they only had the one paper on this stuff, not several. Anyway the paper is astro-ph/0603162 , see section IV where they do precisely what you had in mind and end up with a formula

z = z_doppler + z_grav

where the two z's are written in terms of cosmological parameters. Their derivation is valid for dust dominated Universes only, but should be able to be generalized I would think.

 

[jonmtkisco]

Hi Wallace,
Thanks for finding the Gron and Elgaroy paper. I'm a little surprised at how "conservative" their perspective is. They want to attribute cosmological redshift to the stretching of wavelength due to the expansion of space, despite pointing out the problems that have been identified with that physical description. Oh well, I'm more interested in their specific results than in how they characterize them.

G&E attribute the z = z_gravity + z_doppler formulation to Bondi (1947). They say this is an approximation which is valid only at small fractions of the Hubble radius. There must be some way to generalize the result to larger radii as you say, but apparently no one has done it yet.

I had thought that the gravitational redshift component of the equation would be "positive", in the sense that it would contribute redshift rather than blueshift. Because the past worldline of a photon from a distant source passes through regions of smoothly decreasing density over time, which, in G&E's terminology, would suggest that the photon has climbed "uphill" out of a gravitational well, i.e. redshifted. However, G&E uses the Shell Theorem/Gauss' Law approach to describe the photon's worldline as "falling downhill" due to the gravitational acceleration towards the observer at the coordinate origin, i.e. blueshifted.

I've attached a copy of a chart from Davis & Lineweaver's 10/03 paper illustrating the relative effects of SR Doppler redshift, GR cosmological redshift, and Classical Doppler redshift. It show that, relative to any given "actual" recession velocity, the SR Doppler equation calculates the highest amount of redshift. The GR cosmological redshift equation for an Omega(m)=1, Lambda=0 is blueshifted relative to the SR Doppler equation. The "empty" universe model is further blueshifted from that. And the Classical Doppler equation is the most blueshifted of the three.

It seems that the degree of redshifting in each respective scenario reflects the degree of time dilation in that scenario. Time dilation = 0 for the Classical Doppler scenario which is least redshifted.

It seems contrary to G&E's analysis that the Omega_M=1, Lambda=0 scenario is more redshifted than the empty universe scenario. In other words, adding matter to the universe causes more redshift, indicating that in the "matter" scenario the photon's past worldline has indeed climbed "uphill" rather than falling "downhill." That outcome is more consistent with my description: climbing out of a gravitational well as the temporal/spatial density decreases.

Unless this is just another one of those situations where the weird Milne model confuses the result.

Jon

 

[Wallace]

G&E attribute the z = z_gravity + z_doppler formulation to Bondi (1947). They say this is an approximation which is valid only at small fractions of the Hubble radius. There must be some way to generalize the result to larger radii as you say, but apparently no one has done it yet.

Hmm, I didn't read the fine print. I've never read the back half of that paper in much detail, just vaguely remembered that redshift split thing. This formula in that case may well be equivalent to Whitings, since the derivation of it is very Newtonian in spirit. Maybe it can be generalised to a full GR formula, but then any maybe it just doesn't work. As I said earlier, GR is non-linear so you can't always add things in the way that might seem intuitive. That's quite a sweeping statement though, it may well be possible in this case.

I had thought that the gravitational redshift component of the equation would be "positive", in the sense that it would contribute redshift rather than blueshift. Because the past worldline of a photon from a distant source passes through regions of smoothly decreasing density over time, which, in G&E's terminology, would suggest that the photon has climbed "uphill" out of a gravitational well, i.e. redshifted. However, G&E uses the Shell Theorem/Gauss' Law approach to describe the photon's worldline as "falling downhill" due to the gravitational acceleration towards the observer at the coordinate origin, i.e. blueshifted.

Hmm I would have expected the gravitational component should be 'positive' or blue-shifting. Consider a static space with a varying density. The redshift of a photon depends on the difference in potential between two places which will be an integral over the gravitational force due to the density. There is no direct relationship between the density at two places and the redshift in a static space, so I don't know that this would be expected in a temporal sense either? Obviously in familiar potential wells like Galaxies the density does drop with radius so the 'mover to lower density get redshifted' rule of thumb works, but if you imagine a static potential well that for some reason got more dense with radius the relationship wouldn't hold. What would stay the same is that the photon gets redshifted by climbing out of the well.

I've attached a copy of a chart from Davis & Lineweaver's 10/03 paper illustrating the relative effects of SR Doppler redshift, GR cosmological redshift, and Classical Doppler redshift. It show that, relative to any given "actual" recession velocity, the SR Doppler equation calculates the highest amount of redshift. The GR cosmological redshift equation for an Omega(m)=1, Lambda=0 is blueshifted relative to the SR Doppler equation. The "empty" universe model is further blueshifted from that. And the Classical Doppler equation is the most blueshifted of the three.

It seems that the degree of redshifting in each respective scenario reflects the degree of time dilation in that scenario. Time dilation = 0 for the Classical Doppler scenario which is least redshifted.

It seems contrary to G&E's analysis that the Omega_M=1, Lambda=0 scenario is more redshifted than the empty universe scenario. In other words, adding matter to the universe causes more redshift, indicating that in the "matter" scenario the photon's past worldline has indeed climbed "uphill" rather than falling "downhill." That outcome is more consistent with my description: climbing out of a gravitational well as the temporal/spatial density decreases.

Unless this is just another one of those situations where the weird Milne model confuses the result.

Jon

Something is up. I doubt either G&E or D&L made such a serious error as to give the conflicting results you are suggesting. One or other would have to be very very wrong for this to hold. I'm not sure of the context of the D&L plot, can you post the link to the paper? Without the actual caption it isn't clear what they are getting at with that plot.

 

[yuiop]

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.

Hi again,

I have taken on board your comment that the empty model is not a good fit to the observed data (although strangely it is a better fit than the mass dominated model). I have given this some more thought and think I now have a better grasp on the matter.

Please look at the attached digrams which illustrate co-moving distances in the FRW metric and compare them with Special Relativistic subluminal distances similar to the Milne model.

In the first diagram is a hypothetical universe where the mass content of the galaxies is almost zero and provides a good simple starting point. The left part of the diagram shows the classic "tear drop" curve of light as it travels towards us against the Hubble flow. The path of three receding galaxies are shown (solid diagonal red, blue and green lines) with redshift of 2, 1 and 0.5 respectively. The dashed lines show the corresponding nominal velocities of z equal to the redshift value but this is not an actual physical velocity. The "corrected velocities" are still superluminal in this model for redshift greater than z=1.7. On the right is the same three galaxies plotted on a coordinate system that has been stretched out as described by Ned Wright here http://www.astro.ucla.edu/~wright/cosmo_03.htm by " dividing the spatial coordinate by a(t)". The left part of the diagram clearly shows the lack of gravity in the model as the galaxies follow parallel paths that remain at constant distance from each other. What is surprising about the right diagram is that despite the lack of gravity, light from distant galaxies appears to slowing down as it aproaches us, initially supeluminal and finally c locally. It is surprising because this co-moving model has no gravity! The curved light path is exactly what you would expect to see if you were deep in a greavitational well. This shows that comoving coordinates introduce a fictitious gravitational field even when there is no mass. I suspect that in trying to compensate for the fictitious gravity that is an artifact of the comoving coordinates that the cosmological constant representing repulsive gravity has to be introduced to make the comoving model math the observation that the universe is essentially flat on large scales. Anyway, the distance AD is the "co-moving distance" that is quoted in cosmological calculators and is preferred to the corresponding distance AB as it closer to distances measured as angular or luminosity distances after suitable corrections of 1/(1+z) and (1+z)^2 respectively. Because there is no gravity in this particular model the comoving distance is equal to EF and FG but that is just coincidence because the galaxies are following straight paths rather than curved paths.

In the second diagram, the left part represents subluminal motion and distances in Special Relativistic coordinates compared to the superluminal comoving distances on the right. It is easy to see that the SR distances A'B' C'D' E'F' are all considerably shorter than the comving distances AB, CD, EF in the comoving model and this is one reason that the comoving model is preferred over the SR model because the comoving distances are closer to what is observered. However, the SR model should not be dismissed so lightly. The third diagram shows the SR coordinates with the effect of rapid inflation in the early epoch of the universe added in. The inflation is represented by the flat truncated base of the universal light cone. Conventional time as measured by the decay of radioactive atoms and the formation of large scale structures starts at time O and not at time O', because matter is not in the form of atoms during the inflation period. The dashed circle is just there to illustrate that the distance of a galaxy receding at redshift z=1 has been set to the same value by a suitable choice of inflation radius. By including inflation the distances of galaxies with z<1 is greater (E'F'>EF) and less luminous than the empty model and the distances of galaxies with z>1 is less (A'B<AB) and more luminous than in the empty model (without inflation). This is the effect that is currently explained by an initially collapsing universe that starts to expand at an accelerating rate in more recent times. Although gravity has not been introduced into the model yet this hints that when inflation is taken into account the universe can be modeled largely using SR with some modification for gravity and GR, in a way that matches observation without requiring superluminal motion, expanding space or dark energy accelerating the expansion.

 

[jonmtkisco]

The Davis & Lineweaver http://arxiv.org/abs/astro-ph/0310808" is "Expanding Confusion: common misconceptions of cosmological horizons and the superluminal expansion of the universe" 11/03.

Jon

 

[Wallace]

Hi again,

I have taken on board your comment that the empty model is not a good fit to the observed data (although strangely it is a better fit than the mass dominated model). I have given this some more thought and think I now have a better grasp on the matter.

Why is this strange? The empty universe fits better than matter dominated because in a universe with matter and dark energy the attractive and repulsive parts roughly cancel each other for a period making the Universe looks closer to the empty model. A matter only Universe only decelerates, so is a worse fit than the empty Universe.

What is surprising about the right diagram is that despite the lack of gravity, light from distant galaxies appears to slowing down as it aproaches us, initially supeluminal and finally c locally. It is surprising because this co-moving model has no gravity! The curved light path is exactly what you would expect to see if you were deep in a greavitational well. This shows that comoving coordinates introduce a fictitious gravitational field even when there is no mass. I suspect that in trying to compensate for the fictitious gravity that is an artifact of the comoving coordinates that the cosmological constant representing repulsive gravity has to be introduced to make the comoving model math the observation that the universe is essentially flat on large scales.

In a sense the FRW co-ordinates do introduce an apparent gravity, in that in GR gravity is curvature. In very rough terms the co-ordinate transformations that get you from a flat Minkowski metric to the equivalent FRW metric of the empty Universe curve the spatial and temporal parts individually, such that the overall curvature remains zero. As I say, I am speaking very loosely here, but is essence that is what is done. This means if you take surfaces of constant FRW time, as you have done, you find that the surface is curved, hence there is an apparent gravity.

But, and this is a very important but, none of this matters when assessing the correctness of the model. Regardless of which co-ordinate system you use you will find the same underlying physical model. There is no way in which this 'apparent' gravity can be mistaken for real gravity and spoof results. Co-ordinate transformations do not change observable quantities. This is how your diagrams can mislead you. What you need to do is work out when distance (e.g. luminosity distance) vs redshift curves you get given different physics. Using either FRW co-ordinates or Minkowski co-ordinates you will get the same curves, because these curves relate to things you would observe, and these are not dependent on co-ordinates.

It is very easy in GR to mistake properties of co-ordinates with observations, but it is an important distinction.

this is one reason that the comoving model is preferred over the SR model because the comoving distances are closer to what is observered. However, the SR model should not be dismissed so lightly.

There is no such thing as 'the co-moving model'. There are co-moving co-ordinates, but the model is General Relativity. For an empty universe, the co-moving co-ordinates description of the GR model is exactly equivalent to SR. For a non-empty universe there are 'SR like' co-ordinates (as defined in recent papers by Chodorowski and Lewis et al) as well as the usual co-moving co-ordinates. But these are most definitely not different models, again when related to observables they give exactly the same results. They are just different co-ordinate descriptions of the same models.

I don't really follow you diagrams, but as I say if you want to discriminate between models you need to work out the model predict for what we can observe, which is almost always something as a function of redshift. The actual distances to a given redshift are not measured but can be derived once the model is set via the something vs redshift data. As it stands I think you plots are interesting, but can't be used in the way you are trying to, since they are in terms of arbitrary unmeasurable quantities. To select the best model it is crucial that the models be related to what can be observed.

 

[jonmtkisco]

Hi Wallace,

I doubt either G&E or D&L made such a serious error as to give the conflicting results you are suggesting.

Unless you come up with a better explanation, my guess is that it's just a typo in Davis & Lineweaver Fig 3. I think the labels on the (0,0) and (1,0) lines were inadvertently switched. The chart makes good sense with those two labels reversed.

Hmm I would have expected the gravitational component should be 'positive' or blue-shifting.

OK, I think your reasoning and prediction are persuasive, so I will set aside my idea about the declining cosmic density acting as a gravity well. Blueshift it is.

I want to try out a simpler way to think about gravitational redshift. The standard explanation makes it sound as if gravity acts directly on the energy of a photon, either decreasing the energy (redshift) if it is "climbing out of a gravity well," or increasing the energy (blueshift) if its "falling into a gravity well." To me, it seems easier if one does NOT think of gravity acting on the energy level. Instead, the only effect of gravity is to cause time dilation or contraction.

Thus, if a photon travels from the surface of a massive planet to an observer at a distant point in empty space, the local clock on the planet runs slower relative to the space clock. The observer's faster pace of life (faster clock) causes him to perceive the photon as having a lower frequency (fewer oscillations per clock tick) than would be calculated using the planet clock. The photon has not "changed" at all, it is just perceived differently depending on which clock is used to measure its frequency. This description is consistent with the concept that gravitational redshift does not "occur enroute", instead it "occurs as a single discrete event" at whatever point its frequency is measured. It also is consistent with the fact that in a matter-only universe the Integrated Sachs-Wolfe effect does not affect the redshift of photon which passes through a succession of overdense and underdense regions along its path. The gravitational redshift is calculated solely by the relative clock differential at the two end points. E.g., by comparing the clock at the observer with the clock at the emitter using the SR time dilation equation.

With this narrative in hand, maybe we can separate cosmological redshift cleanly into a velocity component and a clock component as follows.

1. Velocity Component. The equation for this is:

\frac{ \left( V_{emit} + V_{rec} \right) }{2}

where V_{emit} is the proper recession velocity between the emitter and receiver at the time of emission, and V_{rec} is the proper recession velocity between them at the time of reception. The Velocity Component follows the Classical Doppler effect equation v=cz.

2. Clock Component. The equation for this is:

\frac{ \left( \Delta T_{emit} + \Delta T_{rec} \right) }{2}

where \Delta T_{emit} is the relative clock rate difference due to the proper velocity between the emitter and receiver at the time of emission (V_{emit}), and \Delta T_{rec} is the relative clock rate difference due to the proper recession velocity between them at the time of reception (V_{rec}). The Time Component follows the SR time dilation formula.

These equations account for the gravitational decrease in the receiver's proper recession velocity (relative to the emitter) which occurs between the time the photon is emitted and the time it is received. The equations divide by two in order to exclude the gravitational decrease in the emitter's proper recession velocity occurring place after the photon is emitted.

I've attached a diagram showing the proper distance and elapsed time over which (V_{rec}) is calculated at the time of reception.

Jon

 

[Wallace]

Hi Wallace,

Unless you come up with a better explanation, my guess is that it's just a typo in Davis & Lineweaver Fig 3. I think the labels on the (0,0) and (1,0) lines were inadvertently switched. The chart makes good sense with those two labels reversed.

I think your right, I've been scratching my head over that plot and couldn't work out what they were on about. If it's just a typo then I agree it makes sense once the labels are switched.

I want to try out a simpler way to think about gravitational redshift. The standard explanation makes it sound as if gravity acts directly on the energy of a photon, either decreasing the energy (redshift) if it is "climbing out of a gravity well," or increasing the energy (blueshift) if its "falling into a gravity well." To me, it seems easier if one does NOT think of gravity acting on the energy level. Instead, the only effect of gravity is to cause time dilation or contraction.

This is consistent with the standard explanation, since redshift is a measure of the time dilation between the observed and emitted frames. Redshift if you like is the way we can measure time dilation.

Thus, if a photon travels from the surface of a massive planet to an observer at a distant point in empty space, the local clock on the planet runs slower relative to the space clock. The observer's faster pace of life (faster clock) causes him to perceive the photon as having a lower frequency (fewer oscillations per clock tick) than would be calculated using the planet clock. The photon has not "changed" at all, it is just perceived differently depending on which clock is used to measure its frequency. This description is consistent with the concept that gravitational redshift does not "occur enroute", instead it "occurs as a single discrete event" at whatever point its frequency is measured. It also is consistent with the fact that in a matter-only universe the Integrated Sachs-Wolfe effect does not affect the redshift of photon which passes through a succession of overdense and underdense regions along its path. The gravitational redshift is calculated solely by the relative clock differential at the two end points. E.g., by comparing the clock at the observer with the clock at the emitter using the SR time dilation equation.

Yep, redshift and time dilation are always equivalent.

With this narrative in hand, maybe we can separate cosmological redshift cleanly into a velocity component and a clock component as follows.

1. Velocity Component. The equation for this is:

\frac{ \left( V_{emit} + V_{rec} \right) }{2}

where V_{emit} is the proper recession velocity between the emitter and receiver at the time of emission, and V_{rec} is the proper recession velocity between them at the time of reception. The Velocity Component follows the Classical Doppler effect equation v=cz.

I don't quite follow this? How can the recession velocity 'at time of reception' of the distant galaxy be a factor? Once the photon leaves the emitting galaxy it doesn't know or care about what that galaxy does. Unless I'm missing the point of that expression? It would help if you could make it clearer what that expression refers to, i.e. what exactly is it that equals that expression?

2. Clock Component. The equation for this is:

\frac{ \left( \Delta T_{emit} + \Delta T_{rec} \right) }{2}

where \Delta T_{emit} is the relative clock rate difference due to the proper velocity between the emitter and receiver at the time of emission (V_{emit}), and \Delta T_{rec} is the relative clock rate difference due to the proper recession velocity between them at the time of reception (V_{rec}). The Time Component follows the SR time dilation formula.

These equations account for the gravitational decrease in the receiver's proper recession velocity (relative to the emitter) which occurs between the time the photon is emitted and the time it is received. The equations divide by two in order to exclude the gravitational decrease in the emitter's proper recession velocity occurring place after the photon is emitted.

I've attached a diagram showing the proper distance and elapsed time over which (V_{rec}) is calculated at the time of reception.

Jon

Hmm, I'm still confused. Maybe if you demonstrate an example of using this to calculate a redshift it would help?

 

[yuiop]

...

Unless you come up with a better explanation, my guess is that it's just a typo in Davis & Lineweaver Fig 3. I think the labels on the (0,0) and (1,0) lines were inadvertently switched. The chart makes good sense with those two labels reversed.

I think the Davis & lineweaver diagram is correct without the switch (No typo). The (0,0) model has no gravity and no antigravity while the (1,0) model has significant gravity due to normal matter and no antigravity. The reason the velocity for a given redshift is lower in the (1,0) model is due to the slow down of the recessional velocity by gravity which eventually leads to a collapse of the universe. The greater redshift in the (1,0) model relative to the (0,0) model for a given recessional velocity is due to gravitational redshift which is present in the former but not in the latter.

...
Thus, if a photon travels from the surface of a massive planet to an observer at a distant point in empty space, the local clock on the planet runs slower relative to the space clock. The observer's faster pace of life (faster clock) causes him to perceive the photon as having a lower frequency (fewer oscillations per clock tick) than would be calculated using the planet clock. The photon has not "changed" at all, it is just perceived differently depending on which clock is used to measure its frequency. This description is consistent with the concept that gravitational redshift does not "occur enroute", instead it "occurs as a single discrete event" at whatever point its frequency is measured. It also is consistent with the fact that in a matter-only universe the Integrated Sachs-Wolfe effect does not affect the redshift of photon which passes through a succession of overdense and underdense regions along its path. The gravitational redshift is calculated solely by the relative clock differential at the two end points. E.g., by comparing the clock at the observer with the clock at the emitter using the SR time dilation equation.

Your basic concept that the redshift of a photon emitted in an early epoch of the universe is partly due to the reduction in the energy density of the universe during its travel time is good one and I think it is also correct to say it is equivalent to the redshift of a photon climbing out of a gravity well. However, it is synonomous to the concept of the redshift being due to the "stretching" of the photon wavelength by the expansion of space. In other words you can calculate the increase in wavelength using either concept/model and get the same results but it is important to realize they are essentially the same thing and not to apply the both concepts at the same time and double count. Mathematically they are the same, just the interpretation of "what is really happening" is different.

That is one component of the redshift. The other component in comoving coordinates is classic doppler redshift due to recessional velocity relative to the observer at the time of emission. There is no SR time dilation component in comoving coordinates. In the SR model there is a component due to SR time dilation calculated using the subluminal SR velocity with an additional component due to classic doppler redshift due to the SR recession velocity and a component due to gravity but there is no component due to "expansion of space" in SR coordinates. The spacetime in SR coordinates is static and does not admit a cosmological constant or acceleration of expansion by an antigravity effect. I tried to show geometrically in an earlier post that lack of an anti gravity effect in the SR model can be compensated by assuming a (large) component due to rapid inflation right at the start. When I get more time I will try to show what I am talking about in mathematical terms rather than visual geometrical terms. Hopefully I can produce some equations that give the same numerical results as the cosmological calculators and people might find that more convincing. You should be able to do the same the equations you are trying to produce for the components of cosmological redshift. That is the acid test ;)

 

[jonmtkisco]

Hi Wallace,

I don't quite follow this? How can the recession velocity 'at time of reception' of the distant galaxy be a factor? Once the photon leaves the emitting galaxy it doesn't know or care about what that galaxy does.

Apparently the explanation of this point wasn't clear in my post. I said:

The equations divide by two in order to exclude the gravitational decrease in the emitter's proper recession velocity occurring place after the photon is emitted.

The reason for the attached diagram was to show that the equation must take account of the receiver's recessionary movement (away from the emission point) after the time of emission, but must NOT take account of the emitter's recessionary movement (away from the emission point) after that time. We are in agreement that what happens to the emitter after emission isn't relevant. A subtlety is required in order for the equation to accomplish this result from the data available, which after all does not distinguish between the recession of one party and the recession of the other party, since their recession is relative as between each other only. That's why I attribute 1/2 of the total post-emission recessionary movement to each party, rather than attribute the entire recessionary movement arbitrarily to just one of the parties.

Jon

 

[MeJennifer]

I want to try out a simpler way to think about gravitational redshift. The standard explanation makes it sound as if gravity acts directly on the energy of a photon, either decreasing the energy (redshift) if it is "climbing out of a gravity well," or increasing the energy (blueshift) if its "falling into a gravity well." To me, it seems easier if one does NOT think of gravity acting on the energy level. Instead, the only effect of gravity is to cause time dilation or contraction.

Thus, if a photon travels from the surface of a massive planet to an observer at a distant point in empty space, the local clock on the planet runs slower relative to the space clock. The observer's faster pace of life (faster clock) causes him to perceive the photon as having a lower frequency (fewer oscillations per clock tick) than would be calculated using the planet clock. The photon has not "changed" at all, it is just perceived differently depending on which clock is used to measure its frequency. This description is consistent with the concept that gravitational redshift does not "occur enroute", instead it "occurs as a single discrete event" at whatever point its frequency is measured. It also is consistent with the fact that in a matter-only universe the Integrated Sachs-Wolfe effect does not affect the redshift of photon which passes through a succession of overdense and underdense regions along its path. The gravitational redshift is calculated solely by the relative clock differential at the two end points. E.g., by comparing the clock at the observer with the clock at the emitter using the SR time dilation equation.

That is the correct description. Curved spacetime does not influence the frequency of a photon, and neither does it stretches its wavelength as these would be in clear contradiction to the equivalence principle.

The reception of a "red shifted" photon is really a unaltered photon observed by a relatively blue shifted
receiver.

See for instance: http://xxx.lanl.gov/abs/physics/9907017

 

[jonmtkisco]

Hi Kev,

I think the Davis & lineweaver diagram is correct without the switch (No typo). The (0,0) model has no gravity and no antigravity while the (1,0) model has significant gravity due to normal matter and no antigravity. The reason the velocity for a given redshift is lower in the (1,0) model is due to the slow down of the recessional velocity by gravity which eventually leads to a collapse of the universe. The greater redshift in the (1,0) model relative to the (0,0) model for a given recessional velocity is due to gravitational redshift which is present in the former but not in the latter.

I hear ya', but I still think it's a typo. In proper distance coordinates, cosmological redshift in a (0,0) universe should be close to SR Doppler redshift alone. Actually it seems like it should identical to SR Doppler redshift alone, I'm not sure why it's even a separate line. Maybe some sort of coordinate confusion.

Conversely, the higher Omega_m is (with Lambda=0), the more the proper velocity of the observer away from the original emission point slows down during the travel of the photon; therefore the less redshift he will observe as a function of whatever the proper recession velocity was at emission time.

The issue here is not what the "true recession velocity" is; rather we are looking for the relationship between the "true recession velocity" and the "observed recession velocity." So for this specific purpose it doesn't matter whether the overall recession velocity of the universe is relatively high or low. By the way a (1,0) universe will not ever collapse, instead its expansion rate will decrease asymptotically toward zero.

In other words you can calculate the increase in wavelength using either concept/model and get the same results but it is important to realize they are essentially the same thing and not to apply the both concepts at the same time and double count. Mathematically they are the same, just the interpretation of "what is really happening" is different.

Agreed. Double counting must be avoided. The reason I prefer the "clock differential" model of gravitational redshift is because I think it makes it easier to treat the Doppler and gravitational components of redshift in a consistent way.

There is no SR time dilation component in comoving coordinates.

Comoving coordinates sometimes obscure the simplicity or subtlety of what gravity is doing. That's why my discussion is all in proper distance coordinates.

Jon

 

[yuiop]

Hi Kev,

I hear ya', but I still think it's a typo. In proper distance coordinates, cosmological redshift in a (0,0) universe should be close to SR Doppler redshift alone. Actually it seems like it should identical to SR Doppler redshift alone, I'm not sure why it's even a separate line. Maybe some sort of coordinate confusion.

SR distance coordinates are proper distance coordinates. The Davis & lineweaver diagram shows the (0,0) velocity using the FRW metric which uses comoving coordinates which is the accepted way of doing cosmology. As I was trying to show in post #85, there is a fictitious acceleration or curvature even when there is no mass or cosmological constant in the FRW metric using comoving coordinates. The acceleration of the universe that is described as the "discovery of the century" may turn out to be the "blunder of the century" (again) because it is a fictitious force introduced to cancel out the fictitious gravitational effect of the FRW metric. It is a bit like centrifugal force. This is a fictitious force which in Newtonian physics was canceled by the force of gravity of a free falling body. Einstien showed that physics based on two fictitious forces cancelling each other out is not good physics and his breakthrough in GR is that there is no force of gravity acting on free falling bodies.

...
By the way a (1,0) universe will not ever collapse, instead its expansion rate will decrease asymptotically toward zero.

I accept that correction but it should be recognized that there is an implicit assumption of recessional velocity in that statement. Without galaxies having outward escape velocity even a (1,0) universe would collpase.

...
Comoving coordinates sometimes obscure the simplicity or subtlety of what gravity is doing. That's why my discussion is all in proper distance coordinates.

Jon

I agree, but you should bear in mind that all conventional texts on cosmology work in co-moving coordinates.

 

[Wallace]

The reason for the attached diagram was to show that the equation must take account of the receiver's recessionary movement (away from the emission point) after the time of emission, but must NOT take account of the emitter's recessionary movement (away from the emission point) after that time. We are in agreement that what happens to the emitter after emission isn't relevant. A subtlety is required in order for the equation to accomplish this result from the data available, which after all does not distinguish between the recession of one party and the recession of the other party, since their recession is relative as between each other only. That's why I attribute 1/2 of the total post-emission recessionary movement to each party, rather than attribute the entire recessionary movement arbitrarily to just one of the parties.

Jon

(emph mine)

The data doesn't tell us anything about the recession velocity. Only the redshift. I still have no idea how you could use what you describe to calculated the expected redshift, given some other observable in some model.

 

[jonmtkisco]

Hi Wallace,
If you start with a known Hubble rate and proper distance, I was suggesting you could calculate a predicted cosmological redshift. First you calculate the relative proper velocity between (1) the inertial frame of the receiver at reception time, and (2) the inertial frame of the emitter at emission time. Then you just use this calculated velocity to calculate the SR relativistic Doppler shift. I think that should give the correct answer for the cosmological redshift.

My thought was that this calculated velocity already includes gravity's effect in reducing the relative recession velocity during the photon's travel, so you don't need to calculate any separate component for gravitational redshift.

Jon

 

[Wallace]

But does it work?

 

[jonmtkisco]

Hi Wallace,

First, I must clarify that in my simplified description for a "bottoms-up" method to calculate cosmological redshift, I omitted the step which corrects for change in time dilation. The complete set of steps is:

1. Calculate the relative proper velocity between the inertial frames of (i) the receiver at reception time and (ii) the emitter at emission time. In a flat universe, the higher the average background matter density during the travel period, the more this calculated velocity will be reduced, as compared to the relative velocity between the emitter and receiver at the time of emission.

2. Calculate the Doppler redshift using the SR relativistic Doppler redshift equation. The greater the relative recession velocity between the receiver is from the emitter, the more the predicted redshift will be increased by the relativistic correction to the Classical Doppler formula.

3. Calculate the difference in gravitational time dilation beween the inertial frames of (i) the receiver at reception time and (ii) the emitter at emission time. E.g., for an emitter currently at z=1, the scale factor (a) was half of today's value, and therefore the gravitational density was r³ or 8 times greater. Apply the time dilation change as a correction to the relativistic Doppler redshift calculation. The greater the difference in time dilation, the more the total predicted cosmological redshift will be reduced by this correction.

But does it work?

I don't know. I may need some help getting the math straight.

When you ask "does it work", I think you mean, does it calculate the same answer as the cosmological redshift equation, which simply compares the scale factor (a) at the time of reception and emission. It's not immediately apparent that it does, because my method applies two relativistic corrections, and the cosmological redshift equation applies none. But the two relativistic corrections I use affect the calculation in opposite directions, so at least to some extent they tend toward cancelling each other out.

I note that the cosmological redshift equation takes gravity into account, since the gravitational slowing of the increase in (a) over time directly affects the calculated redshift. The gravitational reduction in recession velocity over time is directly proportional to distance, which allows Hubble's law (net recession velocity is proportional to distance) to remain exactly true at the time of emission, reception, and at every time in between. The cosmological redshift equation simply excludes ALL SR relativistic effects: both the relativistic components of the Doppler effect and gravitational time dilation. In this sense the nature of the cosmological redshift equation is entirely Newtonian.

As far as I can see, there are only three possible ways that the cosmological redshift equation can ignore relativistic effects and still be correct: (1) if space itself really is expanding and actually causes wavelength to stretch with the scale factor; (2) if Special Relativistic effects inherently become reduced (asymptotically toward zero) over very large cosmological travel distances, or (3) if the relativistic effects (and the "divide by two" component) of my method all exactly cancel each other out.

I am not inclined to accept #1 above, and #2 is a radical idea that has no apparent explanation. So I hope #3 turns out to be true.

Jon

 

[yuiop]

When you ask "does it work", I think you mean, does it calculate the same answer as the cosmological redshift equation, which simply compares the scale factor (a) at the time of reception and emission. It's not immediately apparent that it does, because my method applies two relativistic corrections, and the cosmological redshift equation applies none. But the two relativistic corrections I use affect the calculation in opposite directions, so at least to some extent they tend toward cancelling each other out.

A simple first test of your equations is to do some numerical calculations and compare them with the results from these cosmology calculators: http://nedwww.ipac.caltech.edu/help/cosmology_calc.html

That will at least tell you if you are in the right ball park.

 

[jonmtkisco]

I do not think that the method I described for a "bottoms-up" calculation of predicted cosmological redshift is consistent with the FLRW metric, so it must be incorrect. In particular, the change in matter density as a function of time does not cause any clock rate differential in the homogeneous FLRW metric. In normalized units, the FLRW metric can be simply written as:

ds² = -dt² + a²(t)(dx² + dy² + dz²)

The cosmic clock (t) is invariant for purely comoving observers as a function of the declining matter density. The cosmic clock is just the timelike spacetime distance orthogonal to a hypersurface of constant comoving physical distance, so:

ds² = -dt².

So in the same way that the declining cosmic matter density does not create any gravitational redshift, it also does not create any clock differential between the emitter and receiver.

A.B. Whiting seems to have been on the right track when he derived the gravitational component of cosmological redshift by calculating the difference between the matter density now and zero matter density. I think the remaining step needed to extend his analysis into a general equation for cosmological redshift is to perform an integration of the SR Doppler redshift at each point between the emitter and receiver, multiplied by an integration of the gravitational redshift at each point between the emitter and receiver (calculated using the matter density now and a matter density of zero):

\frac{\lambda_{r}}{\lambda_{e}} = \int\begin{array}{cc} v^{e}\\v_{r} \end{array} SR \ Doppler \ redshift \\\ \int\begin{array}{cc} \rho^{r} \\ \rho_{0} \end{array} gravitational \ redshift

As Whiting says, just multiplying the SR Doppler redshift and the gravitational redshift calculates the correct instantaneous cosmological redshift for a flat FLRW universe with static density.

Something along these lines is needed so that we can obviate the need for the tradititional explanation that the "expansion of space" physically stretches the wavelength of transiting photons. As regards observational predictions of GR, a model universe where space does not expand must be identical to those of a universe with expanding space. Then we can attribute cosmological redshift simply to the difference between an SR universe (i) without gravity and therefore with a single global reference frame, and (ii) with gravity, and therefore with an infinitude of different local reference frames.

Jon