"The universe is not expanding" — not a joke, but the claim of a real paper published in 2013
In Episode 1 we acquired the tool \(\tilde g=\Omega^2g\) — the operation that swaps the marks on your ruler at every point. Let us go straight to the most extreme use of it. We are going to erase the expansion of the universe entirely. Where does the erased part go? Into the atoms. What this episode shows is that the picture "the universe is not expanding; atoms are shrinking instead" is in no conflict whatever with observation — and that it exists as a serious paper.
When we say "the galaxies are receding", we are measuring a distance. And what do we measure distance with?
Metres. And what is one metre? As bonus episode 2 of the previous series discussed, it is now defined as the distance light travels in \(1/299792458\) of a second. And one second? 9192631770 oscillations of the light emitted by a caesium atom. In the end, every ruler bottoms out in an atom.
This ratio grows with time — that is the most accurate statement of the observational fact we call "the universe is expanding". There is no way to measure numerator and denominator separately. Outside the atom there exists no ruler unrelated to atoms.
There are two reasons a ratio can grow.
No observation distinguishes ① from ②, even in principle. If the ratio is all you can measure, then equal ratios mean identical situations.
And it is exactly a conformal transformation that realises ②. Take Episode 1's \(\Omega\) to be \(1/a(t)\) (with \(a\) the scale factor) and the expansion divides out cleanly and vanishes. What moves in its place is mass.
Mass grows in proportion to \(a\). Since the Bohr radius is \(\hbar/(m c\alpha)\), atoms shrink as \(1/a\).
Why mass moves this way we derive from the equations in Episode 3 (the mass term can keep its form in no other way). For now, take the result: whatever we erased from the expansion has moved, in full, onto mass.
At the left end the galaxies huddle together; at the right end the atom swells. The two look completely different, yet "galaxy spacing ÷ atom radius = 0.500" does not move at all. That number is the only thing a telescope can read.
Someone has taken "the universe is not expanding" seriously. Christof Wetterich, 2013. The title is exactly that: "A Universe without expansion". The opening of the abstract, in translation:
This is not merely a thought experiment. Wetterich builds a concrete model: a scalar field called the cosmon, similar to the Higgs, from which all particle masses arise; its potential is responsible for both inflation and today's dark energy. And he writes: "Our model is compatible with all present observations."
Of course it is. There is no way for it to conflict. If the only measurable thing is a ratio, then every model that reproduces the ratio gets through.
This is the question that must be bothering you. If nothing is expanding, why is the light from distant galaxies red?
① The expanding picture
Space stretches, so the wavelength of light in flight stretches with it. By arrival it has been drawn out by a factor \(1/a\).
② The shrinking-atom picture
Space does not stretch, so the light arrives exactly as it left, unchanged. What changed is the receiving end — the hydrogen atom in the laboratory is heavier than it used to be. The frequency an atom emits is roughly proportional to its mass, so
$$\frac{\nu_{\text{light that arrived}}}{\nu_{\text{today's laboratory hydrogen}}}=\frac{\tilde m(\text{then})}{\tilde m(\text{now})}=a$$That is \(1+z=1/a\) — exactly the same equation as ①. Not "the light stretched" but "the reference grew", and the spectrograph reads the same number either way.
The previous series offered a third way of saying it: "light used to be faster". Side by side:
| Picture | Explanation of the redshift | \(1+z\) |
|---|---|---|
| ① Space stretches | The wavelength of light in flight is drawn out | \(1/a\) |
| ② Light slows down | Light used to be faster; the drop is the reddening | \(1/a\) |
| ③ Atoms shrink | The light is unchanged; the laboratory reference grew | \(1/a\) |
All three give the same number. What this series does over ten episodes is to understand, from the root, why these three lines agree.
Let us be straight. The answer is no. And Wetterich says so himself, in the same abstract. Its last three sentences:
"Other, equivalent choices of field variables" — in other words, he states from the outset that this is a change of variables. The expanding picture, the shrinking picture and the static picture are different ways of writing one and the same theory. So the claim is not "the universe is really not expanding". It is "whether the universe expands is a matter of how you write it".
And that is precisely the theme of this series. The motto once more — quantities with units are bookkeeping; only dimensionless ones are physics. "Distance between galaxies" and "atom radius" both carry units, so both are bookkeeping. Only their ratio is physics.
If the physics does not change, is there any point? Three of them.
What this episode establishes is only that the two pictures are observationally indistinguishable. It does not say Wetterich's model is correct — indeed the physical content of his model (the shape of the cosmon potential, the account of inflation and dark energy) is a genuine hypothesis to be tested independently of the rewriting. The rewriting part is free; the model part is not.
Also, "atoms shrink" is a statement relative to the comoving lattice. Atoms are not shrinking inside your room. Locally an atom is the same size today as yesterday — the phrase only means anything once the comparison partner is a galactic scale. This is exactly the same caveat as the previous series' "the locally measured speed of light is always \(c_0\)".
Since every ruler bottoms out in an atom, the only thing a telescope can measure is the dimensionless ratio "distance between galaxies ÷ radius of an atom". That ratio can grow because the numerator grew (expansion) or because the denominator shrank (atoms shrinking) — either is fine. Use the conformal transformation \(\Omega=1/a\) and the expansion vanishes cleanly; in its place mass grows as \(\tilde m=am\) and atoms shrink as \(1/a\). The redshift comes out of the same equation \(1+z=1/a\) — the light did not stretch, the laboratory reference grew.
This is the claim of a real paper (Wetterich 2013). But as he writes in the abstract himself — "there exist other, equivalent choices of field variables" — it is not new physics but a change of variables. Rewriting is still worth it: the shape of questions like the singularity changes (Episode 6), the arithmetic can get easier, and above all it trains you to decide by procedure what is really physics. Build the dimensionless ratio. If it moves, physics; if not, bookkeeping. That single move carries us to the finale.
Print / PDF: ⌘+P (Ctrl+P on Windows). On screen the slider moves between the two pictures and shows that the ratio does not change. "Show answer" opens the solutions.