What exactly did "the speed of light decreases" come out of? Pinning it down in a single equation
"Cosmology That Clicks" ran on a single auxiliary line from Episode 1 to the finale: read "the universe is expanding" as "the speed of light is slowly decreasing". Bonus episode 4 listed two conditions for that rereading to hold — \(c_B\cdot a=\text{const}\) and invariance of \(\alpha\). The answer was right, but why those two was never made clear. With the conformal transformation prepared in Episodes 1 and 2, both come out of one and the same operation. And the identity of "light slowing down" becomes sayable in a single line.
As we saw in Episode 1, the conformal (Weyl) transformation fits on one line. At every point of spacetime, multiply the ruler of length by \(\Omega\) — that is all.
This is not a coordinate transformation (relabelling the same map). A coordinate change leaves the shape of spacetime alone; a conformal transformation stretches length itself. But it stretches equally in all directions at each point, so angles are unchanged — hence "conformal", preserving form.
And as we stressed in Episode 1, the operation can equally be read not as "we changed spacetime" but as "we swapped the ruler". A change of units, carried out point by point. Moving back and forth between these two readings is the theme of the whole series.
Set the stage. The metric of a flat expanding (FLRW) universe, written with a lattice that does not stretch (comoving coordinates \(x\)) and a scale factor \(a(t)\), is
$$ds^2 = -c_0^2\,dt^2 + a(t)^2\,dx^2$$and the previous series' \(c\cdot t=\text{const}\), once unmasked, was \(a(t)=t/t_0\) (linear expansion). We now carry it, by conformal transformation, to a completely flat spacetime. Two moves suffice.
A coordinate change only, to begin with. Use a new time \(\eta\) whose ticks are the old ones divided by \(a\).
Definition
$$d\eta = \frac{dt}{a(t)} = \frac{t_0}{t}\,dt$$Integrating
$$\eta = t_0\ln\!\frac{t}{t_0}\qquad\Longleftrightarrow\qquad t = t_0e^{\eta/t_0},\quad a(\eta)=e^{\eta/t_0}$$Something interesting has happened. Expansion that was "linear" in cosmic time, \(a\propto t\), becomes exponential in conformal time, \(a\propto e^{\eta/t_0}\). And the metric takes a form with a single overall factor pulled out front.
Inside the brackets nothing expands any more. It is plain Minkowski (flat) spacetime. The entire expansion has been pushed into the one coefficient \(e^{2\eta/t_0}\) sitting in front.
Once a single factor has been pulled out, it is the conformal transformation's turn. Choose \(\Omega = 1/a = e^{-\eta/t_0}\) and it divides out exactly.
Exact Minkowski spacetime. Not a trace of expansion is left. Up to here this is textbook procedure — the most direct way of confirming that FLRW spacetime is conformally flat.
The \(\tilde g\) we just built is written in the time coordinate \(\eta\). But a metric is the same metric no matter which coordinates you write it in. Let us go back to cosmic time \(t\). All it takes is substituting \(d\eta = (t_0/t)\,dt\).
Substitute
$$-c_0^2\,d\eta^2 = -c_0^2\Big(\frac{t_0}{t}\Big)^2 dt^2 = -\Big(\frac{c_0t_0}{t}\Big)^2 dt^2$$so that
$$d\tilde s^2 = -\Big(\frac{c_0t_0}{t}\Big)^2 dt^2 + dx^2$$That coefficient should look familiar. It is the one and only rule from Episode 1 of the previous series.
Put them side by side and the answer is already there.
"The speed of light decreases" was the conformally transformed metric, read in cosmic-time coordinates.
One and the same \(\tilde g\): read it in \(\eta\) and the speed of light is constant; read it in \(t\) and the speed of light falls off as \(c_0t_0/t\). The conformal transformation does the job of erasing the expansion, and the choice of time coordinate decides where the price is paid. The picture the previous series called "easy to understand" and kept using was the product of this two-stage move.
As you turn the knob to the right, the marks near the bottom (the early universe) are stretched out more and more, and with them the path of light straightens. Pushing the expansion into the time direction means literally that stretching. And at the far right, once the stretching is complete (conformal time), light travels at the same speed from beginning to end.
So far this has been about the metric — the container. A conformal transformation acts on matter too. As we saw in Episode 2, under \(\tilde g=\Omega^2g\) mass transforms as
$$\tilde m = \Omega^{-1}m = a\,m$$(this is the only way the mass term \(\int\sqrt{-g}\,m\bar\psi\psi\) can keep its form). Since \(a=t/t_0\) for \(c\cdot t=\text{const}\) —
If mass grows then the Bohr radius \(\hbar/(m c\alpha)\) shrinks — the universe is not expanding; the atoms are shrinking instead. That is Wetterich's picture from Episode 2. And for \(c\cdot t=\text{const}\) the shrinking takes the simplest possible form: proportional to \(1/t\).
Apply the same transformation to temperature and something more startling happens. Temperature is an energy, so \(\tilde T = aT\). But in an expanding universe \(T\propto 1/a\), hence
$$\tilde T = a\cdot\frac{{\rm const}}{a} = \text{constant}$$The transformed universe does not cool. The cosmic microwave background is 2.725 K in every era. So where did the "cooling of the universe", used throughout Episodes 7, 9 and 10 of the previous series, go? It moved onto the growth of mass. Rather than the temperature dropping until binding energies overtake it, the binding energies grow until they overtake the temperature. The same crossing, seen from the other side.
If this is "the same physics", the dimensionless ratios must agree. Recombination is governed by "Rydberg ÷ thermal energy", so let us compute it in both pictures and compare.
The expanding picture (standard)
$$\frac{13.6\ \mathrm{eV}}{k_B\!\cdot\!(2.725\times1101\,\mathrm{K})} = \frac{13.6}{0.2586\ \mathrm{eV}} = 52.6$$The conformally transformed picture (constant temperature, growing mass)
$$\frac{13.6/1101\ \mathrm{eV}}{k_B\!\cdot\!2.725\,\mathrm{K}} = \frac{0.01235}{2.348\times10^{-4}\ \mathrm{eV}} = 52.6$$Numerator and denominator are both off by a factor 1101, and only the ratio survives. That is what "observation does not change" means.
Bonus episode 4 of the previous series listed two conditions for the equivalence to hold. In the language of the conformal transformation, they are not two separate demands.
Condition 1 (\(c_B\cdot a=\text{const}\)) is nothing but the choice \(\Omega=1/a\) itself. It merely writes out what the transformation is.
Condition 2 (\(\alpha\) invariant) is the interesting one. The ingredients of \(\alpha=e^2/4\pi\varepsilon_0\hbar c\) — \(e\), \(\hbar\) and \(c\) — none of them move under a conformal transformation. The electromagnetic action \(\int\sqrt{-g}\,F_{\mu\nu}F^{\mu\nu}\) is conformally invariant in four dimensions and the \(\Omega\)s cancel completely. Therefore
So invariance of \(\alpha\) is not a condition you must protect; it is a theorem that holds by itself once the conformal transformation is done properly. Bonus episode 4 was right to list two conditions — but the second was a consequence of the first.
| ① Space stretches | ② Light slows down | ③ Masses grow | |
|---|---|---|---|
| Metric | The original \(g\) | The conformally transformed \(\tilde g=g/a^2\) (the same object) | |
| Time coordinate | Cosmic time \(t\) | Cosmic time \(t\) | Conformal time \(\eta\) |
| Speed of light | \(c_0\), constant | \(c_0t_0/t\) | \(c_0\), constant |
| Mass | constant | \(\propto t\) | \(\propto t\) |
| Temperature | \(\propto 1/t\) | constant | constant |
| Account of the redshift | Wavelengths stretch | Light used to be faster | Atoms used to be lighter |
| \(1+z\) | \(a_0/a_e=t_0/t_e\) | \(c(t_e)/c(t_0)=t_0/t_e\) | \(\tilde m_0/\tilde m_e=t_0/t_e\) |
All three share the last row. ②and ③ are sisters — the same metric read in different time coordinates, while ① sits exactly one conformal transformation away. The two records the previous series called "the LP and the CD" turn out to be three.
A conformal transformation is a rewriting. Rewriting alone produces no new prediction. So this episode does not claim that \(c\cdot t=\text{const}\) is correct — it says only that if \(a\propto t\), then this is what it looks like. Whether the expansion really goes as \(a\propto t\) is a separate matter, settled by observation.
And so far the answer to that separate matter is unkind. Run \(a\propto t\) back into the early universe at face value and nucleosynthesis breaks: the cooling is too slow (Lewis et al. 2016 find a helium mass fraction of order \(10^{-3}\) against the observed 0.25). The bonus episode takes that head on, but note: even with the picture split three ways by the conformal transformation, the verdict is the same in all three — because dimensionless ratios take the same value in every one of them.
Move the \(c\cdot t=\text{const}\) universe to conformal time \(\eta=t_0\ln(t/t_0)\), then apply the conformal transformation \(\Omega=1/a\), and you get exact Minkowski spacetime. Expansion vanishes without trace. Read that same metric back in cosmic time \(t\) and you have \(d\tilde s^2=-c_B(t)^2dt^2+dx^2\) with \(c_B=c_0t_0/t\) — that is what "the speed of light decreases" really is. The conformal transformation erases the expansion; the choice of time coordinate decides where the price is charged.
On the matter side, masses grow \(\propto t\) and the temperature becomes constant: cooling turns into the growth of mass. And because \(e\), \(\hbar\) and \(c\) all sit still under a conformal transformation, \(\alpha\) is invariant without any condition being imposed — bonus episode 4's "condition 2" was in fact a consequence of "condition 1". What VSL stumbled over was failing to make its operation a conformal transformation at all.
Print / PDF: ⌘+P (Ctrl+P on Windows). On screen the slider moves the time coordinate continuously from cosmic time to conformal time. "Show answer" opens the solutions.