# How to interpret the velocity from relativistic Doppler effect/equation for the redshift of cosmic background radiation? [closed]

This question has a moderation and downvotes issue: One of moderators keeps deleting one last single comment with the crucial piece of information, that clearly explains, why the given answer is wrong

Value and calculation of redshift of the microwave background radiation: https://lambda.gsfc.nasa.gov/education/graphic_history/microwaves.html https://thecuriousastronomer.wordpress.com/2015/07/30/what-is-the-redshift-of-the-cosmic-microwave-background-cmb/
We substitute it into the Doppler formula for light and determine the velocity from it.

What and in relation to what is this velocity? From this velocity, time dilation and length contraction can be determined. In relation to what?

In ordinary Doppler, we distinguish between the velocity of the emitter and the receiver. In relativistic Doppler we do not have such distinction, because the emitter and the receiver can have a velocity defined only relative to each other, so it's the same for them both, but in opposite direction.

What could be both a source and a receiver at a redshift of factor 1100?

In our reference frame - Is there a redshift of light, that cannot reach us?

The Special Relativistic Doppler effect formula should not be used to interpret cosmological redshifts. Mandatory reading is section 3.1 of Davis & Lineweaver (2003).

Cosmological redshifts are due to the emission of a photon in one frame of reference and the receipt of that photon in an entirely different frame of reference at a later cosmic epoch. They are caused by the expansion of space, not the movement through space. They can be interpreted in terms of recession velocities but only approximately obey the Special Relativistic formula at low redshifts

Hubble's law is the proportionality between proper distance and rate of change of proper distance. The proper distance to where the CMB was released is about 46 billion light years - the exact value depending on the adopted cosmological parameters. This gives an implied recession velocity of $$3.2$$ times the speed of light. Clearly this value wouldn't be possible using the special relativistic formula.

Using Hubble's law with light travel time distance yields a velocity that is roughly $$c$$ multiplied by the look back time as a fraction of the age of the universe. That is because $$H_0^{-1}$$ is roughly the age of the universe. It (a velocity calculated in this way) has no particular physical meaning.

The plot below is from wikipedia showing the relationship between redshift and recession velocity: The redshift of the CMB is about 1100.

If you do choose to interpret the velocity from the Doppler shift formula, then that is the relative velocity between the gas that emitted the radiation then to us now (see Bunn & Hogg 2009). However, this is not the velocity given by Hubble's law, which, I reiterate, relates rate of change of proper distance to the proper distance - i.e. the separation of two objects at the same epoch.

In your edited question you ask a further question - the measured redshift could be arbitrarily high. In practice we will not observe light with redshifts higher than 1100 because the universe was opaque to radiation prior to the epoch when that was emitted. However, in principle we might measure gravitational waves with higher redshifts from the earlier universe.

• I disagree physics.stackexchange.com/a/738016 no more comments to this answer, if moderators don't delete this one. Commented Sep 17, 2023 at 7:38

It is the velocity of Earth relative to a huge spherical shell of what was plasma back when the CMB was emitted. That shell was just part of a roughly uniform plasma which was, on average, not moving relative to the average distribution of matter in the universe; that average distribution picks out one reference frame among all the others. It is called the co-moving frame. The reason one can pick out such a frame is that the matter is distributed with close to homogeneous density on the largest scales and close to homogeneous velocity distribution also (i.e. isotropic with regard to any chosen centre). There is evidence however of small inhomogeneities even at very large distance scales and this impacts on some of the interpretation of evidence of the dynamics of the expansion.

For the second part of the question, regarding the frequency shift, the notion of a relative speed of source and detector is not needed. Rather, what you want is the calculation of momentum conservation along a null geodesic. The momentum conservation follows from the field equation when the metric has a certain form. The light follows a null geodesic. The frequency shift is the difference between what is observed by observers fixed relative to the co-moving frame at different spacetime locations. Another way to think of it is to think of two null geodesics following the worldllines of successive wavefronts. These two geodesics draw apart along with, and indeed in exact proportion to, the universal expansion.

• Thank you. I know. And here is the second part of this question: physics.stackexchange.com/q/737460 Commented Nov 20, 2022 at 9:44
• Thank you, my saviour. Nonetheless, you've read the rest :) What now? Commented Nov 20, 2022 at 12:08
• "It is the velocity of Earth relative to a huge spherical shell of what was plasma back when the CMB was emitted." - This depends on how you define the relative velocity, but it's correct if you (reasonably) define it by parallel transporting along the light cone. See also What does general relativity say about the relative velocities of objects that are far away from one another?
– Sten
Commented Nov 20, 2022 at 13:27
• @Sten Only now do I know what I got myself into here. Commented Aug 8, 2023 at 6:32

ProfRob's answer and Andrew Steane's answer are mutually exclusive. The former is wrong, the latter is right, and here is why.

In this case, Doppler gives us current velocity and Hubble gives us current velocity for the current distance, because comoving distance and proper distance are defined to be equal at the present time.

That's the source, given by ProfRob himself: https://en.wikipedia.org/wiki/Comoving_and_proper_distances

If the comoving distance to a galaxy is denoted $$\chi$$, the proper distance $$d(t)$$ at an arbitrary time $$t$$ is simply given by

$$d(t)=a(t)\chi$$

The proper distance $$d(t)$$ between two galaxies at time $$t$$ is just the distance that would be measured by rulers between them at that time.

For the current value of the scale factor $$a(t)=1$$ we have $$d(t)=\chi$$. This means, that Doppler equation, that gives us the relative velocity between two co-moving frames at the co-moving distance, is currently appropriate for the proper distance as well, because they are defined as equal at the present time.

See for yourself. Calculate the velocity from the Doppler equation for the current CMB redshift and compare it with the velocity given by the Hubble's Law for the current value of Hubble constant (even assuming different, inconsistent measurements) and the distance equal to $$ct_0$$ where $$t_0$$ is the current age of the universe. In his recently edited answer, ProfRob argues, that Using Hubble's law with light travel time distance yields a velocity that is roughly $$c$$ multiplied by the look back time as a fraction of the age of the universe. - Not At All. It yields $$c \pm 0.03\ c$$. If you want ProfRob, you can multiply it by $$1$$, not a fraction of the age of the universe. Calculate it. In addition, he explains his miscalculation by the reciprocal of Hubble constant not being exactly equal to the universe age and concludes, that therefore this velocity makes no physical sense. Not only that explanation makes no sense, but it's based on miscalculation. The interpretation of the result of proper calculation of this velocity is here.

ProfRob's start point are the values $$46.5$$ billion light years and $$3.2\ c$$ and his whole further argumentation is based on them. I'm questioning them in my other question, that is also given below with wider description. It was deleted, so I give you this instead and I hope the original will be undeleted.

On the one hand ProfRob argues, that I can't use the Doppler, because of general relativity, superluminal expansion with $$3.2\ c$$ (Doppler's velocity is limited to $$(-c, c)$$) and the fact, that the cosmological redshift is caused by the expansion, not the movement through space. Below I'm explaining, why I use the Doppler precisely because of the expansion and not the movement through space. On the other hand he writes, that if I do choose to interpret the velocity from the Doppler shift formula, then that is the relative velocity between the gas that emitted the radiation then to us now. Then and Now are two different, cosmic epochs, separated in time roughly by the age of the universe. That doesn't change the fact, that Doppler gives us current velocity. So "general relativity" and Doppler can't be both right in this case.

The argument about then and now actually applies to the case with the accelerated movement of the emission source through the non-expanding space. At the moment of emission, there was a redshift $$z_e$$ due to the relative velocity $$v_{relE}$$ of the source and the receiver at the moment. I'm introducing a shifted wavelength at the moment of emission: $$\lambda_{e}=(1+z_e)\lambda_0$$. Receiver gets $$\lambda_r=(1+z_r)\lambda_e$$ for the Doppler eq. with $$v_{relR}$$, that would be the relative velocity, if the emitter was moving with the constant speed from the moment of emission. If that was the case with expanding space, ProfRob would be right, but it's not. In the expanding space, photon's wavelength changes like the photon itself was a piece of a rubber band, stretched between the emitter and the receiver. That's why its redshift gives us the current, relative velocity from the Doppler eq.

In my opinion, the source of inconsistency between general relativity (which is supposedly a basis of calculations giving $$46.5\ GLy$$ and $$3.2\ c$$ values) and the Doppler is in calling the integration-based calculated recession velocity (as well as the proper size of the universe) a part of GR. Friedmann equations, as a solution of Einstein's equations, are a part of GR. Friedmann–Lemaitre–Robertson–Walker metric is a part of GR. Explicit form of the scale factor, derived from the Friedmann equations, is a part of GR. In my opinion, what is not a part of GR, is the integration of the generic metric that gives the current, proper distance. This metric equation uses the explicit form of the scale factor derived from GR and may be valid for every single spacetime frame, but if you integrate it, you get the Expanding confusion: Time integral of the reciprocal of the scaling factor from Hubble parameter equation. It was deleted, so I give you this instead and I hope the original will be undeleted.

I'd also like to quote Davis & Lineweaver (2003) given by ProfRob.

The relationship between proper time at the emitter and proper time at the observer is thus

$$\Delta t=\Delta t_0 \gamma (1+v/c)$$
$$\quad =\Delta t_0 \sqrt{\frac{1+v/c}{1-v/c}}$$
$$\quad=\Delta t_0 (1+z)$$

This is identical to the GR time dilation equation. Therefore using time dilation to distinguish between GR and SR expansion is impossible.

The determinant of our time dilation relative to the emission source frame at the moment of emission is redshift.

$$\frac{\Delta t}{\Delta t_0}=(1+z)$$
$$(1+z)$$ is equal to the Doppler value $$\sqrt{\frac{1+v/c}{1-v/c}}$$, so we can get the velocity from it.

There is one more piece of a puzzle. It seems that even ProfRob didn't read, what he recommended: Bunn & Hogg 2009. Please, find equations $$(5)$$ and $$(6)$$, read the description and compare it with the above calculation.

These are the conclusions at the end of chapter III:

We do not expect to have convinced purists who refuse even to talk about $$v_{rel}$$ to change their minds. They would dismiss $$v_{rel}$$ as a mere "coordinate velocity," not an "actual velocity," and take no interest in it. If purists have the courage of their convictions, they would say that it makes no sense to try to attach labels such as "Doppler" or "gravitational" to the observed redshift. This position is unassailable, and we have no wish to argue against it. We do claim, however, that if you wish to try to talk about $$v_{rel}$$, then the definition proposed in this section is the most natural way to do so. Because this definition of $$v_{rel}$$ results in the Doppler formula entirely explaining the redshift, we claim that you should either be a purist and refuse to try to label the cosmological redshift, or you should label it a Doppler shift.

And here is the calculation of the time left to the of the world:

$$t_v=13.77\pm0.059\ \cdot 10^9\ y$$
$$v=0.99999835 \pm 0.00000??\ c$$
$$t_c$$ : time of reaching the speed of light in our co-moving frame by the place of emission of CMB photons, that reach us today
$$H_0\approxeq H(t_v)\approxeq H(t_c)$$ (current expansion is roughly exponential and $$v$$ is almost equal to $$c$$)

$$v = H_0 \cdot d_v = H_0 \cdot c t_v$$
$$H_0 = \frac{1}{t_v} \frac{v}{c}$$

$$c = H_0 \cdot d_c = H_0 \cdot ct_c$$
$$t_c= \frac{1}{H_0} = t_v \frac{c}{v}$$

$$\Delta t = t_c - t_v = (\frac{c}{v} - 1)\ t_v =$$ ;)

You may calculate the velocity given by the Hubble's law for the current value of the Hubble constant and the distance $$ct_v$$, and compare it with the given $$v$$ from the Doppler equation.