# Redshift of distant galaxies: why not a doppler effect?

How can I explain to my 17 year old pupils that the observed redshift of distant galaxies cannot be interpreted as a doppler effect and inescapably leads to the conclusion that space itself is expanding?

I understand that this redshift is well explained in general relativity (GR) by assuming that space itself is expanding. As a consequence, distant galaxies recede from us and the wavelength of the light is "streched". Expansion, redshift and the Hubble law are explained coherently in GR, as well as many other phenomena (e.g. the cosmic microwave background), and the GR predictions about redshift agree with observations.

I understand that the redshift of distant galaxies cannot be explained as a doppler effect of their motion through space. Why exactly is a pupil's doppler interpretation wrong?

My first answer: "Blueshifted galaxies (e.g. Andromeda) are only seen in our local neighborhood, not far away. All distant galaxies show a redshift. At larger distances (as measured e.g. with Cepheïds) the redshift is larger. For a doppler interpretation of the redshift distant galaxies we must necessarily assume that we are in a special place, to the discomfort of Copernicus. In this view, space cannot be homogeneous and isotropic." Is this answer correct?

My second answer: "A doppler effect only occurs at the moment the light is emitted, whereas the cosmological redshift in GR grows while the light is traveling to us." My problem with this answer (if it is correct): what observational evidence do we have for a gradual (GR) increase of the redshift, disproving the possibility of an "instantaneous doppler shift at the moment of emission"?

My third answer: "For galaxies at $z>1$ you can only have $v<c$ if you use the doppler formula from special relativity (SR): $v=\frac{(z+1)^2-1}{(z+1)^2+1}\cdot c$". My problem with this answer: what's wrong with using the doppler formula from SR as long as someone views the universe as static, in a steady state? With just the right amount of dark energy to balance the gravitational contraction, if you wish?

My fourth answer: "Recent observations of distant SN Ia show a duration-redshift relation that can only be explained with time dilation [see Davis and Lineweaver, 2004, "Expanding Confusion etc."]" My problem with this answer: does time dilation prove we have expanding space, in disagreement with a doppler effect?

My fifth answer would involve the magnitude-redshift relation for distant SN Ia [Davis and Lineweaver], but that's too complicated for my pupils.

• There are always different levels of explanation. While technically not correctly characterized as a doppler redshift, I would not say that explaining it to 17 year olds that way without involving general relativity is exceptionally poor teaching. Even the "special place" Copernican interpretation is not required, unless you want to hold on to a "rigid space" model. I would say that at this level letting go of a rigid space is a good idea and using language like "expanding space makes it look like the galaxies are retreating ever faster and it causes a doppler effect-like redshift" is OK. – CuriousOne Jan 9 '16 at 22:03

My first answer: "Blueshifted galaxies (e.g. Andromeda) are only seen in our local neighborhood, not far away. All distant galaxies show a redshift. At larger distances (as measured e.g. with Cepheïds) the redshift is larger. For a doppler interpretation of the redshift distant galaxies we must necessarily assume that we are in a special place, to the discomfort of Copernicus. In this view, space cannot be homogeneous and isotropic." Is this answer correct?

In other words, it is more likely that we are not in a special place and the universe is expanding than that everything in the universe is flying away from us. This is also supported by the fact that we cannot find anything else particularly special about our location in the universe: the galaxy we're in is typical, the group of galaxies our galaxy is in is typical (if a little low on the mass scale compared to clusters like Virgo or Coma), etc.

My second answer: "A doppler effect only occurs at the moment the light is emitted, whereas the cosmological redshift in GR grows while the light is traveling to us." My problem with this answer (if it is correct): what observational evidence do we have for a gradual (GR) increase of the redshift, disproving the possibility of an "instantaneous doppler shift at the moment of emission"?

We actually do have evidence for this. When the light passes through an especially massive cluster of galaxies on the way to us, the photons will gain energy as the fall into the cluster, and lose energy as they come out. If the universe is static, the photons would gain as much energy as they lose, only being deflected. With the expansion of the universe accelerating, though, the photons gain more energy when they fall into a well than when they come out, because the accelerating expansion of the universe has made the well more shallow while the photon was traveling through it. When this happens to a cosmic microwave background photon, it is known as the integrated Sachs–Wolfe effect.

"My third answer: For galaxies at $$z > 1$$[...]" The exception there is correct. The general Doppler shift in special relativity is given by:

\begin{align} f_r = \frac{1 - \frac{v}{c} \cos\theta_s}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} f_s \end{align} with $$\theta_s$$ the angle between the velocity $$\vec{v}$$ and the line of sight, as measured by the source. If you have a redshift of $$z$$, then your velocity is at least $$v_{\mathrm{min}} = c \frac{(z+1)^2 - 1}{(z+1)^2 + 1},$$ with any value up to $$c$$ allowable for the right choice of $$\theta_c$$. Fun fact: the rapidity $$\phi_{\mathrm{min}}$$ for $$v_{\mathrm{min}}$$ is defined by $$v = c \tanh\phi$$, leading to $$\phi_{\mathrm{min}} = \ln(1 + z)$$.

For what it's worth, we also see the effects of Doppler shifts of galaxy motion on redshifts when we study clusters of galaxies. Wikipedia discusses them in the Redshift-space distortions article. In particular, the "fingers of god" effect causes the redshifts of clusters of galaxies to be elongated along the line of sight, and the "pancakes of god" can elongate redshifts perpendicular to the line of sight.

My fourth answer: "Recent observations of distant SN Ia show a duration-redshift relation that can only be explained with time dilation [see Davis and Lineweaver, 2004, "Expanding Confusion etc."]" My problem with this answer: does time dilation prove we have expanding space, in disagreement with a Doppler effect?

This one does not carry information about whether the Doppler effect is relevant. The stretching out of a signal's wavelengths, with the speed of light held constant, will also cause the duration of the signal to increase, causing an apparent time-dilation. You can play around with slowing down and speeding up audio signals to see how this works - you need to do some extra work to keep the pitch the same if you do that. The converse is also true - if you just alter all of the pitches in an audio signal, you'll alter the duration, too, if you don't do extra work.

You can also throw in the observed existence of the CMB. It is very hard to explain using any model that doesn't have an expanding universe (I say "very hard" because I don't want to preclude the possibility of someone more clever than I figuring out a way in the future).

In Stephen Weinberg's well-known book "The First Three Minutes," he talks solely about Doppler shifts. A good review and analysis of the "expanding space" vs. Doppler shift question was provided by Bunn and Hogg, "The kinematic origin of the cosmological red shift," Am. J. Phys. 77 (8), 2009, pp. 688-694. They make convincing arguments that the red shift is best understood as a series of Doppler shifts in overlapping space-time regions small enough so that Minkowski (flat) space-time geometry is an excellent approximation. Regardless, they say in the Conclusion: "There is no “fact of the matter” about the interpretation of the cosmological redshift: what one concludes depends on one’s coordinate system or method of calculation." Their argument for the Doppler shift says it is more "natural" because it is consistent with a number of well-established facts about the general relativistic theory of gravity.

• While I don't work in this area, I think that the reason many authors discourage people from thinking about cosmological redshifts as "Doppler" is that thinking in terms of "motion away from us" makes several basic errors attractive (think "we're in the middle" and "the universe is expanding into pre-existing space"). Bunn's treatment is sensible but arises from a sophisticated understanding of what "velocity" is involved. – dmckee --- ex-moderator kitten Jan 10 '16 at 2:59
• Yes I also think that cosmological redshift is stressed under this name to emphasise space expansion. However it is not observationally different from a Doppler shift. – Alchimista Nov 18 '17 at 10:36

How can I explain to my 17 year old pupils that the observed redshift of distant galaxies cannot be interpreted as a doppler effect and inescapably leads to the conclusion that space itself is expanding?

You can't. It is a Doppler effect. There's no difference in general relativity between the relative motion of galaxies and any other relative motion.

(A surprising number of people think that there is. Ask them to quantify the difference by defining a tensor field that's nonzero when the space between objects is expanding and zero when the objects are merely moving away from each other. They won't be able to do it.)

There's only one kind of redshift in general relativity. All of the various redshift formulas are special cases of it applying only to certain spacetime geometries and sometimes to certain states of motion. When more than one of the formulas applies, they all give the same answer since they're different descriptions of the same phenomenon. For example:

• Flat spacetime is static in two different ways (the obvious way and Rindler coordinates), and the SR and gravitational redshift formulas agree in each case.
• Minkowski space is an FLRW cosmology in two different ways (the trivial constant-$$a$$ way and the Milne cosmology) and the SR and cosmological redshift formulas agree in each case.
• De Sitter space is static and is FLRW in at least two ways, and the gravitational and cosmological redshift formulas agree.

A general formula for the GR redshift that works in every situation and from which all of the formulas mentioned above can be derived is $$1+z = \frac{\mathbf x\cdot \mathbf v_e}{\mathbf x\cdot \mathbf v_r}$$ where

• $$\mathbf v_e$$ and $$\mathbf v_r$$ are the four-velocities of the emitter and receiver at the time of emission and reception respectively
• $$\mathbf x$$ is a four-vector tangent to the light's path at some point (such as the light's four-momentum, or any nonzero multiple of it)
• The vectors are parallel transported along the light's path to a common location before the dot product is taken

"[...] For a doppler interpretation of the redshift distant galaxies we must necessarily assume that we are in a special place, to the discomfort of Copernicus. [...]"

This isn't correct; it's easy to construct a toy cosmology where Hubble's law applies equally everywhere and the redshift is manifestly a Doppler shift, correctly given by the special relativistic formula. Just let all the galaxies start at a point in Minkowski space and move inertially after that with velocities that are evenly distributed in (four-)velocity space. This is the Milne model (already linked above).

"A doppler effect only occurs at the moment the light is emitted, whereas the cosmological redshift in GR grows while the light is traveling to us."

This isn't correct; there's no generally covariant way to attribute any particular frequency/wavelength change to the light at any particular location on its worldline. Even in SR, you could just as well say that the Doppler shift occurs when the light is absorbed, or partially at emission and partially at absorption, depending on what inertial frame you choose.

"For galaxies at $$z>1$$ you can only have $$v if you use the doppler formula from special relativity [...]". My problem with this answer: what's wrong with using the doppler formula from SR as long as someone views the universe as static, in a steady state? With just the right amount of dark energy to balance the gravitational contraction, if you wish?

You can't use the SR formula in general because it assumes flat spacetime, and in, e.g., Einstein's static universe, spacetime isn't flat.

In the Milne model, spacetime is flat, and the SR formula does work, as long as you're careful to distinguish $$dx/dt$$ Minkowski-coordinate velocities (which don't exceed $$c$$) from $$d\chi/d\tau$$ FLRW-coordinate velocities (which do).

However, the more important point is that when the SR formula doesn't work, it's not because something fundamentally different is going on. The same thing is going on, just not the special case of it to which that formula applies.

• " it's easy to construct a toy cosmology where Hubble's law applies equally everywhere and the redshift is manifestly a Doppler shift" How does such a model work at large distances where Hubble's law predicts $v>c$? – Rob Jeffries Sep 26 '19 at 12:05
• @rob-jeffries Hubble's law applies to the comoving velocity which is based on FLRW coordinates. There are many versions of the SR formula but a lot of them take dx/dt in Minkowski coordinates, so you need to do a coordinate conversion. In this model the FLRW coordinates are essentially polar coordinates on Minkowski space (in the 1+1 dimensional case they're the same as Rindler coordinates with the role of space and time swapped). The comoving velocity turns out to be the SR rapidity (times c), so you could also use a version of the formula that takes rapidities "natively." – benrg Sep 26 '19 at 17:54

There are two parts to understanding this:

1. Space is expanding - we know this to be true because as distance increases, so does recessional velocity (Hubble's law). So the more space in-between, the faster it recedes. Therefore the space itself is responsible for the recessions, and thus must be expanding. There is simply so much evidence for Hubble's law, that the probability that every galaxy (except very close galaxies) just happens to be receding from us is minuscule.

• In the sources rest frame: the light source emits light at the rest wavelength and as the light propagates through space it's physical properties (wavelength) expand with space. N.B. amplitude doesn't increase because it's not physical.
• In the observers rest frame: the source is moving away from us, so when it emits the light is immediately Doppler shifted red-wards, and then proceeds through space at this wavelength.

Note both ways of thinking in 2 fall under the definition of cosmological redshift, despite from the observers frame it looks a lot like a Doppler shift. I think the distinction is that any Doppler shift due to the expansion of space, should be considered cosmological redshift.

Hope this helps.

I think the problem is that the term Doppler Effect is being interpreted in different ways, and that is leading to confusion.

I would explain to your students that a change in the apparent wavelength of light can be caused by three factors, as follows.

1) The source and the observer moving relative to each other in a space that is not expanding.

2) The source and observer being in a space that is expanding.

3) The wavelength being changed by curvature of space.

The distinction between 1) and 2) is subtle, and is hard to pin down both conceptually and experimentally. If you decide to interpret the term 'Doppler effect' as meaning only 1), you then have a problem in explaining 2). It is better, in my view, to say that the frequency is changing as a result of a changing difference between the source of the observer, and then explain how the change it distance comes about.