# The cosmological redshift

Jorrie
Gold Member
One point that the physics literature seems to agree upon without controversy is that that the clocks of fundamental comovers run at the same rate in FRW coordinates. E.g. Peebles textbook p. 59: ...
Sure, but you cannot do 'cosmological Doppler shifts' in pure FRLW coordinates, because fundamental observers are not moving relative to each other - they are at fixed spatial coordinates, with zero coordinate velocity. Also sure, there are no gravitational potential gradients in FRLW coordinates, hence also no gravitational redshifts. To do cosmological Doppler shifts, you have to work in Schwarzschild (or equivalent) coordinates, including both types of time dilation factor, as applicable. Minkowski is just a special case of Schwarzschild. This is essentially what B&H, Peacock and others did.
Second, you seem to be missing the point of his eq 16: The velocity shift is a redshift, and the gravitational shift is a blueshift, so the two factors tend to negate each other. Indeed, his equation can be valid only if the SR time dilation and gravitational time dilation are exactly equal with opposite signs, which results in the emitter and observer sharing the same cosmological time.
If you look at my prior post, you will notice that I did not miss that point. However, contrary to what you state, Peacock's equation 16 is valid for any matter-only universe, flat or not. Velocity- and gravitational time dilation will not cancel out for open or closed models, so the equation will give a different result to your 1+z = 1+v/c. I'm still busy testing eq. 16 in a numerical integration, so I'll comment more fully on it (and the rest of your post) later.

Sure, but you cannot do 'cosmological Doppler shifts' in pure FRLW coordinates, because fundamental observers are not moving relative to each other - they are at fixed spatial coordinates, with zero coordinate velocity.
No, we're getting caught up in semantics here. You are referring to 'comoving coordinates' (see Wikipedia article of that name), which are a specialized form of FRW coordinates. I am referring to FRW proper distance coordinates, which are portrayed in Ned Wright's diagram and similar diagrams by Davis and Lineweaver. In FRW proper distance coordinates, comovers (other than the origin itself) are moving, not stationary, and accumulated Doppler shifts can be discretely calculated.
Also sure, there are no gravitational potential gradients in FRLW coordinates, hence also no gravitational redshifts.
Which means that there is no gravitational time dilation as between any 2 comovers in FRW coordinates -- their clocks run at the same rate.
To do cosmological Doppler shifts, you have to work in Schwarzschild (or equivalent) coordinates, including both types of time dilation factor, as applicable. Minkowski is just a special case of Schwarzschild. This is essentially what B&H, Peacock and others did.
No, I have explained exactly how cosmological Doppler shifts are done in FRW proper distance coordinates, and my approach is mathematically equivalent to Peacock's textbook equation 3.67 which he uses to make the same point as me. Peacock's eq 3.67 does not include any special relativistic elements, so it must be done in FRW proper distance coordinates, not Minkowski.

As I said, 'constant cosmological time' is a feature of Minkowski coordinates and not of FRW coordinates. The whole point of constant cosmological time is to contrast it against the SR time-dilated relationship between Minkowski comovers. That dichotomy is what Ned Wright's and Davis & Lineweaver's charts portray. If you read Peacock's entire section it will be crystal clear that he says constant cosmological time is specifically a feature of FRW coordinates. Davis & Lineweaver make it very clear that constant cosmological time is a unique feature of FRW coordinates, and they emphasize that it is not a feature of Minkowski coordinates.

The B&H article doesn't really get into the subject of constant cosmological time at all. I have already explained in excrutiating detail how B&H get themselves into trouble by not realizing that Minkowski local frames sum to the FRW light travel distance, not the length-contracted Minkowski distance.
However, contrary to what you state, Peacock's equation 16 is valid for any matter-only universe, flat or not.
I did not say that eq 16 was or wasn't valid in any universe containing gravity. I said you should test it and let us know what you find. I did say I agree with Davis & Lineweaver that Peacock was wrong to suggest that the SR Doppler shift formula can be used 'globally' in an empty universe in FRW proper distance coordinates.
Velocity- and gravitational time dilation will not cancel out for open or closed models, so the equation will give a different result to your 1+z = 1+v/c.
The fact that the SR and gravitational time dilation do not exactly cancel out in Minkowski or Schwarzschild coordinates does not mean that they will generate a different cosmological redshift result than my equation. (To the extent I implied otherwise in my last post, that was a misstatment.) Hopefully eq 16 will produce the same redshift result as the standard cosmological formula and as my equation does. (You agree that my equation produces the same result as the standard cosmological formula). With so many different coordinate systems to choose from, there's plenty of room for multiple equations to generate the same correct redshift result. But each such equation can be applied only within its corresponding coordinate system.

Your original premise for introducing eq 16 to the discussion was that if it works in one coordinate system, then my equation must be wrong in a different coordinate system. Hopefully we can agree now that that is not the case.

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Jorrie
Gold Member
No, we're getting caught up in semantics here. You are referring to 'comoving coordinates' (see Wikipedia article of that name), which are a specialized form of FRW coordinates. I am referring to FRW proper distance coordinates, which are portrayed in Ned Wright's diagram and similar diagrams by Davis and Lineweaver.
I don't like getting dragged into these sort of arguments - that's why there are Advisors on this forum. I'll rather spend time on a numerical simulation, since you and me tended to get going forward with that in the past, which is better than going into endless arguments...

Nevertheless, http://www.astro.ucla.edu/~wright/cosmo_02.htm#DH", especially section 2.2 on velocities.
I have explained exactly how cosmological Doppler shifts are done in FRW proper distance coordinates, and my approach is mathematically equivalent to Peacock's textbook equation 3.67 which he uses to make the same point as me. Peacock's eq 3.67 does not include any special relativistic elements, so it must be done in FRW proper distance coordinates, not Minkowski.
Without going into it, in his textbook Prof. Peacock did nothing of the sorts that you are doing. His subsequent 2001 and 2007 (diatribe) papers illuminate his approach very clearly. He uses local Minkowski frames, with a second order gravitational effect worked in for accuracy - that's his 'diatribe' #16 equation (originally mentioned by Ich on this thread, not by me, BTW). Bunn & Hogg leave out the second order correction, which is small in any case. I am using the Peacock approach in my numerical integration of redshifts (which is getting along well, but not finished yet - it's difficult to determine the integration loop error accumulation).
Your original premise for introducing eq 16 to the discussion was that if it works in one coordinate system, then my equation must be wrong in a different coordinate system. Hopefully we can agree now that that is not the case.
No, I did not say your equation is wrong - just that it does not portray Doppler shift accumulation, but rather expansion redshift accumulation. The reason is simple: your treatment is technically wrong from a Doppler shift point of view. I'm not going to argue that further, sorry. Let's rather wait to see results from the simulation.

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Can somebody tell me why we dont see one side of the sun differently than the other since we are travelling away from one side and towards the other? Seems to me that the sun should be phased from red to bright yellow.

sylas
Can somebody tell me why we dont see one side of the sun differently than the other since we are travelling away from one side and towards the other? Seems to me that the sun should be phased from red to bright yellow.
We do see the two sides differently, but the effect is smaller than you describe and hard to measure.

We are actually going around the Sun. The only difference in velocity between one side and other of the Sun is due to the Sun's own rotation... which is about once every 25 days, at the Equator. The solar equatorial radius is about 695000 km. The corresponding velocity at the limb of the sun is up to about 2 km/sec. This is 0.0000067 of the speed of light.

The corresponding redshift between one side and the other works out to be about z= 1.000013

You certainly can't see the difference, but with care it can be measured. And it has been.

See: The Solar Red-Shift, by L.A. Higgs, in Monthly Notices of the Royal Astronomical Society, Vol. 121, p. 421-435, 1960.

Of course the solar redshift also has the gravitational component, but they did also consider the effect you describe.

Cheers -- sylas

I don't like getting dragged into these sort of arguments - that's why there are Advisors on this forum. I'll rather spend time on a numerical simulation, since you and me tended to get going forward with that in the past, which is better than going into endless arguments...
Reluctantly I agree. You are bouncing from critisism to critisism without coming to grips with the fundamental points made by the sources I reference. You seem mostly interested in pushing Peacock's Diatribe eq 16 which is in a different coordinate system, and therefore is not directly relevant to whether my equation and the accompanying explanation in my OP are correct. I suggest you start your own thread about eq 16 so it can have the full discussion it deserves without diverting from this thread.
Nevertheless, http://www.astro.ucla.edu/~wright/cosmo_02.htm#DH" are not in FLRW coordinates - his second spacetime diagram (hyperbolic simultaneity) is in conformal (Minkowski) coordinates, based on proper distance and cosmological time.
Jorrie, I feel like you are looking for opportunities to create confusing about what I said. My reference to Ned was for the very narrow purpose of pointing out one example of FRW proper distance coordinates. Of course I was referring to Wright's top diagram. That's the one that uses FRW proper distance coordinates. Of course his bottom diagram is Minkowski. I already said that multiple times in this thread. Do I have to repeat every detail in every post so that you won't jump on my words?

Ned is showing how the two coordinate systems are different but related. He's showing that constant cosmological time in FRW coordinates transforms into time dilation in Minkowski coordinates. All of which 100% supports my effort to convince you that cosmologists agree, without controversy, that constant cosmological time is a unique feature of FRW coordinates and it is not an inherent feature of time-dilated Minkowski coordinates. Which is the exact opposite of what you claimed a couple of posts ago. If after all this discussion you do not accept that simple, simple, simple point, then we are stalled.
If you haven't done so yet, I recommend that you read the Geraint F. Lewis (et. al) paper http://arxiv.org/pdf/0707.2106v1", especially section 2.2 on velocities.
Been there, done that, adds nothing to the particular topic of this conversation. That paper focuses on conformally flat coordinates, which is yet another of many ways of looking at the expansion. In section 2.2 you referenced, the authors are starting from FRW comoving coordinates, not FRW proper distance coordinates, but in any event they say:

"A fundamental definition of distance in general relativity is the proper distance, defined as the spatial separation between two points along a hypersurface of constant time. Given the form of the FLRW metric (Equation 1), the radial distance from the origin to a coordinate x along a hypersurface of constant t is;

Dp(t) = a(t) x

Taking the derivative with respect to coordinate time [which is synchronous for all comoving observers (fixed x) and is equivalent to their proper time] we obtain what we will refer to as the proper velocity..."
Without going into it, in his textbook Prof. Peacock did nothing of the sorts that you are doing.
How can you make such a careless assertion? Here's what Peacock's textbook, which was written in 1999 and revised several times through at least 2005, says at p.87:

"One way of looking at this issue is to take the rigid point of view tha 1 + z tells us nothing more than how much the universe has expanded since the emission of the photons we now receive. Perhaps more illuminating, however, is to realize that, although the redshift cannot be thought of as a global Doppler shift, it is correct to think of the effect as an accumulation of the infinitesimal Doppler shifts caused by photons passing between fundamental observers separated by a small distance:

$$\frac{dz}{1+z} = \frac{H_{z}dl_{z}}{c}$$ (3.67)

(where dl is a radial increment of proper distance). This expression may be verified by substitution of the standard expressions for Hz and dl/dz."

It is easy to see that Peacock's formula is mathematically equivalent to mine. And that there is no SR time dilation parameter in his equation. And when he refers to a "rigid point of view", clearly he's referring to the same rigid view you hold.
The reason is simple: your treatment is technically wrong from a Doppler shift point of view. I'm not going to argue that further, sorry.
Your unsupported assertion that my treatment is wrong has no persuasive value (but I still respect you!) Sorry to see you go, but this part of the discussion was just going around in circles anyway.

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Ich
nutgeb, you lost me too. I don't know where to start with you re-edited post. There are at least two factual errors, and some major misconceptions, but obviously I don't know how to communicate these points. We've been through it several times, while you would admit now that there is time dilation in Minkowski coordinates, in the same post you say its application is wrong. I just don't know what you're trying to say.

Ich, what don't you understand about this part, which is referring to summing up locally observed light travel distances in Minkowski coordinates?

"This should be obvious: no comoving Milne observer sees the distances to galaxies near his own location to be contracted; only distances to galaxies far away from his location are contracted. When each such Milne observer contributes to the group calculation his own measurement of only the local part of the worldline as he observers it, each such local measurement is non-contracted, and therefore the global sum of all such local measurements cannot be length contracted."

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Jorrie
Gold Member
Trying to avoid the "circles", here is my attempt towards some progress.
It is easy to see that Peacock's formula is mathematically equivalent to mine. And that there is no SR time dilation parameter in his equation.
The equivalence stops once you have replaced $H_{z}dl_{z}$ with the approximation $dv_{z}$. Thereafter you treat it as a classical Newtonian Doppler shift that can be accumulated. Peacock doesn't. He uses his result to show connections to de Broglie wavelength redshifts and expansion in the empty (Milne) model in general. AFAIK, he only returned to redshift proper in his 2001 and 2007 papers (maybe also in others), where he treats them in the standard relativistic sense.

Since the $dl}$ steps are small, your method uses the approximation:

$$1 + dv = 1 + tanh(H dl) \approx 1 + H dl = 1 + H dt = 1 + da/a$$

Chalnoth has shown in reply #3 that an accumulation of (1+da/a) factors yields the correct cosmic redshift ratio (1+z). It then follows that an accumulation of (1+dv) factors will yield an approximately correct result. I think it boils down to a mix of SR and comoving coordinates. Before the approx sign it is Minkowskian and to the right it is cosmological length and time. You get away with it for small steps, where the difference is negligible and the end result looks good.

Maybe it is good enough for all practical purposes - the engineering part of me likes that. The scientific part of me does not. :)

Edit: I think the part nobody here likes is the claim that Peacock, B&H, etc. must be in error when using the full relativistic treatment when they apply the Minkowskian velocity and that the Newtonian equation should be used. To use the left side of the approximate equality above in the integration is not wrong, because the right hand side gives the proper answer from an expansion point of view (and so should the left hand side). IMO, one should just not call it a Doppler shift calculation - it simply mimics the expansion factor accumulation (1+da/a) closely.

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The equivalence stops once you have replaced $H_{z}dl_{z}$ with the approximation $dv_{z}$.
Jorrie, I appreciate your focused response here. But I don't follow you when you say I have replaced an exact result with an approximation in my equation. My $H_{t}$ is in the same units as Peacock's $H_{z}$, and my $d_{t}$ is just a convenient substitution for $d_{l}$ which is in the same units as Peacock's $dl_{z}$. There is no approximation taking place, the parameters are one and the same.
Since the $dl}$ steps are small, your method uses the approximation:

$$1 + dv = 1 + tanh(H dl) \approx 1 + H dl = 1 + H dt = 1 + da/a$$
Ah, I see how you are misinterpreting my equation. You are suggesting that because the $dl}$ steps are small, I must have converted from FRW coordinates to Minkowski coordinates. But I made no such conversion, explicitly or implicitly. There is nothing that prevents one from using FRW coordinates for individual infinitesimally small local frames. The FRW local frames will exactly match local observations. FRW frames scale perfectly from infinity all the way down to zero. Which is what I have done in my equation. (The fact that writers often want to use local Minkowski frames in order to get back to a coordinate system where they can use SR doesn't mean that one is compelled to make the conversion.) And I think what I did is exactly consistent with what Peacock intended in his 3.67.
Thereafter you treat it as a classical Newtonian Doppler shift that can be accumulated. Peacock doesn't. He uses his result to show connections to de Broglie wavelength redshifts and expansion in the empty (Milne) model in general.
There goes that disparaging term "Newtonian" again. What I said that my equation, and therefore Peacock's 3.67 as well, can be interpreted as either the classical Doppler shift formula, or as the SR Doppler shift formula with the time dilation factored out. The characterization is up to the preference of the interpreter. I think you are familiar with the fact that the classical Doppler equation for a moving emitter and stationary observer (1 + v/c) is exactly equal to the SR Doppler equation divided by the SR time dilation factor.

I think its interesting that in his description of 3.67, Peacock used the neutral term "Doppler shift", when it must have been obvious to him that there was no element of time dilation in his equation. My guess is that in the context of a textbook, he did not want to launch into the lengthy explanation that would be required to avoid the predictable knee-jerk reaction that there was something 'non-relativistic' or 'Newtonian' (hold your nose!) if FRW inherently invokes a Doppler shift that includes no element of time dilation.
AFAIK, he only returned to redshift proper in his 2001 and 2007 papers (maybe also in others), where he treats them in the standard relativistic sense.
I'll just reiterate that, by Peacock's own introduction to the Diatribe on his website, it was never intended to be treated like a peer-reviewed paper, it is more in the nature of musings, which he published in a permanent form because they had been referenced by multiple authors in published papers. The use of the word "Diatribe" in the title is a big clue as to how he thought about it. And I think he would be dismayed to have it interpreted as a bolt-in replacement for a section of his textbook. Besides, we already discussed that his Diatribe eq 16 must be in Schwarzschild coordinates while his textbook eq 3.67 is in FRW coordinates, so he could not have meant for one equation to supersede the other; rather the two equations complement each other.
Edit: I think the part nobody here likes is the claim that Peacock, B&H, etc. must be in error when using the full relativistic treatment when they apply the Minkowskian velocity and that the Newtonian equation should be used. To use the left side of the approximate equality above in the integration is not wrong, because the right hand side gives the proper answer from an expansion point of view (and so should the left hand side).
I suppose you intend "full relativistic treatment" to again mean that it's better than a mere "Newtonian" treatment. Sigh. You still need to learn that there is no moral superiority in injecting SR treatments inappropriately into calculations involving FRW coordinates, but obviously you resist being cured of that blind spot.

However, in the interest of being as balanced as possible in the discussion of whether B&H's use of SR is "right" or "wrong", and to hopefully wrap up the discussion of this point with you and Ich, I will soften my conclusion as follows:

= = =

Bunn & Hogg's excellent paper uses parallel transport and a 'world tube' analysis to demonstrate informally that the cosmological redshift can be properly interpreted as an accumlation of local Doppler shifts. B&H caution that their approach includes an element of error in cases where there is any spacetime curvature (i.e., cosmic gravity). The size of this error becomes vanishingly small as the local segment size approaches zero.

However, B&H fail to articulate that their use of the SR velocity addition formula introduces a second element of error even if the spacetime curvature is zero. When the light travel distances measured locally for each segment are summed, they do not exactly equal the FRW 'global' light travel distance, instead they only asymptotically approach it. This is because the SR formula includes an element of time dilation, yet any non-zero amount of time dilation will cause the sum of the local light travel times to diverge from the FRW 'global' light travel distance. This discrepency arises from the well-known fact that the clocks of fundamental comovers in FRW coordinates all run at the same rate, so a photon passing between them experiences no net time dilation. B&H's approach starts with Minkowski segments and tries to aggregate them in to an FRW 'global' worldline. In doing so, it tries to compare parameters from two different coordinate systems without applying the necessary coordinate conversion equation. Fortunately, the size of this second error also becomes vanishingly small as the local segment size approaches zero.

It is not really fair to critisize a short paper because it wasn't expanded to include additional analysis. But B&H did miss an opportunity in their paper to perform a simpler segment analysis arranged the other way around. Instead of summing local Minkowski segments, one starts with the FRW 'global' worldline length and then divides it into an arbitary number of FRW 'local' worldline segments. There is no need to convert these local FRW segments into Minkowski coordinates. The FRW segments nevertheless represent exactly the locally observed segment light travel distance, even for infinitesimal segment lengths. The lengths of these FRW local segments will of course sum exactly to the FRW 'global' worldline length, including at segment sizes > 0, and whether or not spacetime curvature is present. So both kinds of error inherent in the B&H approximation are eliminated. In this pure FRW alternative analysis, the FRW local segments must of course be summed directly without employing the SR velocity addition formula, because the latter would introduce the same mismatch between the collective lengths of the parts and the whole that exists in B&H's approach.

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Jorrie
Gold Member
Jorrie, I appreciate your focused response here.
Thanks. Now, to focus the discussion even more, I have PM'd you a spreadsheet (link) with my solution to the Peacock, B&H, 1/a and your models, aggregated over some 8000 steps, z=0 to 1000. When you have the time, please investigate and let me know if you agree. Especially, my spreadsheet fails to show the "second element of error" that you refer to here:
However, B&H fail to articulate that their use of the SR velocity addition formula introduces a second element of error even if the spacetime curvature is zero. When the light travel distances measured locally for each segment are summed, they do not exactly equal the FRW 'global' light travel distance, instead they only asymptotically approach it. This is because the SR formula includes an element of time dilation, yet any non-zero amount of time dilation will cause the sum of the local light travel times to diverge from the FRW 'global' light travel distance.
Despite your many explanations, I still fail to understand what "FRW 'global' light travel distance" means. AFAIK, there is only one definition for light travel distance: $D_{lt} = c(t_{observe}-t_{emit})$, where t_emit is expressed in the reference frame of the observer. This is essentially the SR distance. Do you have a reference for FRW global light travel distance?

Ich
Ich, what don't you understand about this part, which is referring to summing up locally observed light travel distances in Minkowski coordinates?

"This should be obvious: no comoving Milne observer sees the distances to galaxies near his own location to be contracted; only distances to galaxies far away from his location are contracted. When each such Milne observer contributes to the group calculation his own measurement of only the local part of the worldline as he observers it, each such local measurement is non-contracted, and therefore the global sum of all such local measurements cannot be length contracted."
I understand this part. Do you understand that each observer will actually improve her contribution if he includes time dilation in its (maximum pc ) frame?
I don't know how often I repeated that point: it's actually relativistic doppler shift, but can be approximated as a "classical" shift for small distances.

Despite your many explanations, I still fail to understand what "FRW 'global' light travel distance" means. AFAIK, there is only one definition for light travel distance: $D_{lt} = c(t_{observe}-t_{emit})$, where t_emit is expressed in the reference frame of the observer. This is essentially the SR distance. Do you have a reference for FRW global light travel distance?
Jorrie, the SR 'global' light travel distance is Lorentz-contracted, which is obviously different from the definition you give. It is illustrated for the empty Milne model in Ned Wright's lower diagram, as the vertical distance along the t axis.

The formula you cite is in fact the formula for FRW 'global' light travel distance. It is illustrated for the empty Milne model in Ned Wright's upper diagram, again as the vertical distance along the t axis. It is not Lorentz-contracted; therefore it is longer than its SR counterpart. There is no problem expressing t in the observer's local frame, because as I've said many times, in FRW coordinates t = t', the observer's t is exactly the same as the emitter's t', there is no time dilation as between fundamental comovers.

If you sum up locally observed (SR or FRW) light travel segments, you will automatically calculate (asymptotically) the FRW 'global' light travel distance, not the SR "global light travel distance. As I said in an earlier post, due to the nature of the Lorentz contraction, locally observed segments have asymptotically approaching zero Lorentz contraction. That's why you can't sum up local non-Lorentz-contracted segments to calculate a Lorentz-contracted total.

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Jorrie