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 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.
Answer
Your first answer is the most correct one:
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).
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