c·t = CONST, THAT CLICKS EPISODE 28 ／ Part IV — the theory standing closest to this one

The failure was not moving c, but continuing to call it c

VSL — where the
surgery went wrong "The speed of light was faster in the past" holds two different things.
Cut them apart and the observable content is, entirely, "$\alpha$ varies".

What you need: the particle horizon, logarithms, Episode 19's practice$\Delta\alpha/\alpha=-\Delta c/c$

The second patient of Part IV is VSL (variable speed of light). It is the theory closest to this series, which is exactly why the surgery cuts so well. Inside "the speed of light was faster in the past" there are again two different things — a change of units and a claim that a dimensionless quantity moves. Cut them apart and it becomes clear where VSL's surgery went wrong. The answer first: the failure was not moving $c$, but continuing to call it $c$.

01There are four $c$s to begin with

Before cutting, check what is being cut. The $c$ called "the speed of light" in fact appears four times, in four different roles.

Which $c$	What it does
the $c$ in Maxwell's equations	the propagation speed of electromagnetic waves
the $c$ in the Lorentz transformation	causal structure — the tilt of the light cone
the $c$ in $E=mc^2$	the conversion factor between mass and energy
the $c$ in the Einstein equations	the coupling of curvature to matter, $8\pi G/c^4$

Conclusion of §01

In principle these four are separately movable quantities.
So "$c$ varies" does not say which one varies — the point made by Ellis & Uzan (2005).
There were not two things to operate on but four or more.

02What VSL actually claims is a variation of $\alpha$

VSL (Albrecht & Magueijo 1999, among others) makes the choice explicitly — hold $e$ and $\hbar$ fixed and move $c$. The fine structure constant then comes along.

The dimensionless quantity that follows $$\alpha=\frac{e^2}{4\pi\varepsilon_0\hbar c}\qquad\Longrightarrow\qquad \frac{\Delta\alpha}{\alpha}=-\frac{\Delta c}{c}$$

Conclusion of §02

The observable content of "the speed of light varies" is, entirely, "$\alpha$ varies".
── When Extra 3 of the previous series said "VSL collides with atomic clocks", this one line was the substance.

03How tightly is $\alpha$ pinned?

Following Episode 19's practice, convert to bits — a bound of $10^{-n}$ means $\log_2(10^n)$ bits are pinned down.

Measurement	Epoch	Bound on $\|\Delta\alpha/\alpha\|$	Bits pinned
Laboratory (CODATA 2022)	today	$1.6\times10^{-10}$	32.5 bit
Atomic clocks (13.8 Gyr extrapolation)	$z\simeq0$	$1.4\times10^{-8}$	26.1 bit
Oklo natural reactor	1.8 Gyr ago	$1.1\times10^{-8}$	26.4 bit
Quasar absorption lines	$z\sim2$	$1.0\times10^{-5}$	16.6 bit
CMB	$z=1100$	$4.0\times10^{-3}$	8.0 bit
Nucleosynthesis	$z=4\times10^{8}$	$1.0\times10^{-2}$	6.6 bit

In the recent universe 26 bits are held down, and even at nucleosynthesis 6.6 bits. On Episode 19's scale, this is a quantity known not to move to a precision exceeding Koide's relation (15.7 bits).

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04The heart — how much must $c$ change to solve the horizon problem?

VSL's selling point is solving the horizon problem without inflation. How much change does that take? Rewrite Episode 27's particle horizon for a varying $c$.

Two lines

with $c=c_0(a/a_0)^n$, in the radiation era ($a\propto t^{1/2}$, $dt\propto a\,da$)

$$\chi=\int\frac{c\,dt}{a}\ \propto\ \int a^{n}\,da=\frac{a^{n+1}}{n+1}$$

divergence as $a\to0$ requires

$$\boxed{\ n<-1\ }\qquad(\text{the standard }n=0\text{ gives }\chi\propto a\text{, which does not diverge})$$

So VSL requires that, going back, $c$ grows at least as fast as $1/a$. Since $\alpha\propto1/c\propto a^{-n}$ —

The required variation of $\alpha$ $$\frac{\alpha(z)}{\alpha_0}\ \ge\ 1+z$$

Epoch	$1+z$	Required $\Delta\alpha/\alpha$	Times the observational bound
Oklo (1.8 Gyr ago)	1.14	0.14	$1.3\times10^{7}$×
Quasars ($z\sim2$)	3.0	2.0	$2.0\times10^{5}$×
CMB ($z=1100$)	1101	1100	$2.8\times10^{5}$×
Nucleosynthesis ($z=4\times10^8$)	$4\times10^{8}$	$4\times10^{8}$	$4\times10^{10}$×

The thing this episode most wants to say

Observation excludes solving the horizon problem with a smooth power-law VSL.
At nucleosynthesis the requirement is $4\times10^8$ against a bound of $10^{-2}$ — ten orders of magnitude over.
And even at Oklo, 1.8 billion years ago, it is already seven orders short.

05The VSL that survives — and the price of surviving

Yet VSL is not dead, because Albrecht and Magueijo propose not a power law but a phase transition — $c$ drops abruptly at some moment and is constant thereafter.

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Put the transition before nucleosynthesis and none of §03's bounds applythere is no $\alpha$ data at $z>4\times10^8$

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The horizon problem is still solvedit suffices that $c$ was large enough before the transition ($10^{32}$× if placed at the Planck era)

✗

The price: zero predictions in the observable eraafter nucleosynthesis $\alpha$ is exactly constant — indistinguishable from standard cosmology

Figure: redshift across, $|\Delta\alpha/\alpha|$ up. Grey points are observational bounds, and the region above them is excluded. Move the transition epoch with the slider — the instant it escapes the data, its predictions in the observable era go to zero as well.

Where to put the transition, $\log_{10}(1+z_t)$ 1+z = 1000

$\Delta\alpha/\alpha$ for phase-transition VSL power-law VSL ($n=-1$) observational bounds (above is excluded)

Drag the slider right to make the transition earlier and the violet curve slips under the grey points. But the moment it does, the curve sits on the floor ($\Delta\alpha/\alpha=0$) across every observable epoch. Escaping exclusion and losing predictions were the same operation.

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06The reveal — three fates

	Does $\alpha$ move?	Collision with observation	Predictions
Power-law VSL	yes ($O(1)$)	ten orders over at nucleosynthesis	falsified
Phase-transition VSL	only before nucleosynthesis	none	zero in the observable era
c·t=const (conformal)	exactly invariant	none	notation; zero in itself

The same words, "the speed of light varies", split three ways depending on what you hold fixed. Episode 9's Exercise 5 — "VSL is killed by atomic clocks and this picture is not" — was rows one and three of this table. Today the middle row is added.

Conclusion of §06 — the result of the surgery

The contents of "the speed of light varies" are exactly Episode 3's two ──
(A) a change of units (says nothing) and (B) a claim that a dimensionless quantity moves (observable).
VSL chose (B) but kept the name of (A).
── The right name is "variable $\alpha$ theory".

Where the surgery went wrong VSL did the surgery halfway. It correctly recognised that "$c$ alone is not enough" and fixed $e$ and $\hbar$ — that much matches Episode 3. But it did not change the name. As a result, the phrase "the speed of light varies" hides the content ("$\alpha$ varies"), and the 26 bits of constraint from measurements of $\alpha$ stop being visible head on. The failure was not moving $c$, but continuing to call it $c$.

07Not to be confused — the running of $\alpha$

This is not a time variation $$\alpha^{-1}(0)=137.036\ \longrightarrow\ \alpha^{-1}(M_Z)=127.951\qquad(\text{a difference of }6.6\%)$$

This is the renormalisation-group running seen in Episodes 11 and 14. It is a dependence on energy scale, not a variation in time. Every bound in §03 asks whether $\alpha$, measured at the same energy scale, has moved over cosmic time — a different axis, and confusing them means comparing 6.6% with $10^{-8}$. The same separation Episode 21 made in putting the a-theorem on another axis.

The honest line — what this episode assumes

① "There are four $c$s" is Ellis & Uzan's (2005) account. Which $c$ you vary changes the theory, and this document follows only the choice in which $\alpha$ varies. A VSL that varies the Lorentz-transformation $c$ — that is, the causal structure itself — requires a separate discussion.

② §03's bounds are a summary of representative values. For quasar absorption lines, Webb and collaborators have claimed a significant variation of $\Delta\alpha/\alpha\simeq-0.6\times10^{-5}$, with Keck and VLT disagreeing in sign, and the debate continues. The $10^{-5}$ used here is a conservative summary bound, not a single measurement. The CMB and nucleosynthesis bounds also move by factors of a few depending on how degeneracies with other parameters are handled.

③ The condition $n<-1$ in §04 assumes radiation domination and a power-law $c(a)$. Varying $c$ also changes the equation for $H$, so strictly one must solve a modified Friedmann equation — this is an order-of-magnitude argument looking only at whether the horizon diverges.

④ "Phase-transition VSL has zero predictions in the observable era" concerns $\alpha$. There are formulations giving VSL other predictions (fluctuation spectra, its approach to the flatness problem), which are tested separately — the claim here is the single point that measurements of $\alpha$ cannot distinguish it.

⑤ This episode does not refute VSL. It counts that the power-law version is excluded and that the phase-transition version escapes the $\alpha$ measurements. The purpose of the surgery is to name the comparison hidden inside a name.

Exercises (solvable with this episode's formulas alone)

Why is "the speed of light varies" not a claim by itself?
Show the answer
Because $c$ is dimensionful — bookkeeping — so it means nothing until you say what is held fixed. And $c$ appears in four separate places (Maxwell, the Lorentz transformation, $E=mc^2$, the Einstein equations), so you must also say which $c$.
What does VSL actually claim?
Show the answer
Holding $e$ and $\hbar$ fixed while moving $c$ makes $\alpha=e^2/4\pi\varepsilon_0\hbar c$ move as $\Delta\alpha/\alpha=-\Delta c/c$. The observable content is entirely "$\alpha$ varies" — the right name is "variable $\alpha$ theory".
Find the variation of $c$ needed to solve the horizon problem.
Show the answer
With $c=c_0(a/a_0)^n$, $\chi=\int c\,dt/a\propto\int a^n da=a^{n+1}/(n+1)$. Divergence as $a\to0$ requires $n<-1$, so with $\alpha\propto a^{-n}$, $\alpha(z)/\alpha_0\ge1+z$.
Compare requirement and bound at nucleosynthesis.
Show the answer
At $z=4\times10^8$ the requirement is $\Delta\alpha/\alpha\ge4\times10^8$ against a bound of $10^{-2}$ — a ratio of $4\times10^{10}$, ten orders over, so smooth power-law VSL is excluded. Even Oklo, 1.8 Gyr ago, is already seven orders short.
(Harder) Why is phase-transition VSL not excluded, and at what price?
Show the answer
Confining the variation of $c$ to before nucleosynthesis ($z>4\times10^8$) puts it in a region where no $\alpha$ data exist, so no bound applies. The price is that it has zero predictions for $\alpha$ in the observable era — indistinguishable from standard cosmology. Escaping exclusion and losing predictions are the same operation.

Summary — the failure was not moving $c$

First we checked what was being cut: the $c$ called "the speed of light" appears four times in four roles — Maxwell's equations, the Lorentz transformation, $E=mc^2$, the Einstein equations. So "$c$ varies" does not say which one (Ellis & Uzan 2005).

VSL makes the choice explicitly — hold $e$ and $\hbar$, move $c$. Then $\Delta\alpha/\alpha=-\Delta c/c$, and the observable content is entirely "$\alpha$ varies". And $\alpha$ is pinned to 32.5 bits in the laboratory, 26.4 bits at Oklo 1.8 billion years ago, and 6.6 bits even at nucleosynthesis — on Episode 19's scale, known not to move to a precision beyond Koide's relation.

The heart was §04. Solving the horizon problem requires the particle horizon $\chi\propto a^{n+1}/(n+1)$ to diverge, i.e. $n<-1$, which demands $\alpha(z)/\alpha_0\ge1+z$. At nucleosynthesis that is a requirement of $4\times10^8$ against a bound of $10^{-2}$ — ten orders over. Observation excludes solving the horizon problem with a smooth power-law VSL.

VSL is not dead, though: the phase-transition version confines the change of $c$ to before nucleosynthesis. But escaping exclusion and losing predictions turned out to be the same operation — after nucleosynthesis $\alpha$ is exactly constant and it is indistinguishable from standard cosmology.

And the result of the surgery. The contents of "the speed of light varies" are exactly Episode 3's two — (A) a change of units (says nothing) and (B) a claim that a dimensionless quantity moves (observable). VSL chose (B) but kept the name of (A). The right name is "variable $\alpha$ theory". The failure was not moving $c$, but continuing to call it $c$.

Next — Episode 29 The next patient is MOND. "Below an acceleration of $a_0=1.2\times10^{-10}\ \mathrm{m/s^2}$, Newton's law changes" — and that $a_0$ is dimensionful, so Episode 3's surgery applies directly: small compared with what? Amusingly, when you look for something to make $a_0$ dimensionless with, $cH_0$ is sitting right next door ($a_0/cH_0=0.18$). Identity, coincidence or physics — Episode 19's procedure will sort it. And we watch the question "dark matter or MOND?" turn into Episode 5's balance: which reduces the description length?

This document is Episode 28 of "c·t = const, That Clicks", written for physics-minded high-school and university readers. That $c$ appears in several places in different roles, so that a VSL is undefined until one says which $c$ varies, is the point of Ellis & Uzan (2005, Am. J. Phys. 73, 240). VSL is due to Albrecht & Magueijo (1999, PRD 59, 043516), Barrow (1999) and others. $\alpha=e^2/4\pi\varepsilon_0\hbar c$ and $\Delta\alpha/\alpha=-\Delta c/c$ under fixed $e,\hbar$ follow from the definition. §03's bounds summarise representative values: CODATA 2022's $\alpha^{-1}=137.035999177(21)$, the atomic-clock bound $|\dot\alpha/\alpha|<1.0(1.1)\times10^{-18}$/yr (Lange et al. 2021, PRL 126, 011102), the Oklo natural reactor, quasar absorption lines, the CMB and nucleosynthesis — for quasar absorption lines Webb and collaborators have claimed a significant variation, with Keck and VLT disagreeing in sign, and the debate continues. The $10^{-5}$ used here is a conservative summary bound rather than a single measurement, and the CMB and nucleosynthesis bounds move by factors of a few with the treatment of degeneracies. The relation $\chi\propto a^{n+1}/(n+1)$, the condition $n<-1$, the requirement $\alpha(z)/\alpha_0\ge1+z$ and the excess factors ($4\times10^{10}$ at nucleosynthesis) are computed here (kenshou/calc32.py) — varying $c$ also changes the equation for $H$, so strictly a modified Friedmann equation is needed; this is an order-of-magnitude argument about whether the horizon diverges. "Phase-transition VSL has zero predictions in the observable era" concerns $\alpha$; other formulations give VSL further predictions, tested separately. The running of $\alpha^{-1}$ from 137.036 to 127.951 at $M_Z$ is energy-scale dependence, not time variation (Episodes 11 and 14). This document does not refute VSL: it counts that the power-law version is excluded and that the phase-transition version escapes the $\alpha$ measurements, and names the comparison hidden inside a name. Linear expansion ($c\cdot t=$const, $R_h=ct$) is a minority model under examination. The academic standard remains the $\Lambda$CDM model including inflation. ── To make a PDF, use your browser's Print dialogue (sliders freeze and answers are hidden in the print version).

Print / PDF: ⌘+P (Ctrl+P on Windows). On screen, moving the transition epoch shows predictions vanishing at the instant exclusion is escaped. "Show the answer" opens each solution.

Which \(c\)	What it does
the \(c\) in Maxwell's equations	the propagation speed of electromagnetic waves
the \(c\) in the Lorentz transformation	causal structure — the tilt of the light cone
the \(c\) in \(E=mc^2\)	the conversion factor between mass and energy
the \(c\) in the Einstein equations	the coupling of curvature to matter, \(8\pi G/c^4\)

Epoch	\(1+z\)	Required \(\Delta\alpha/\alpha\)	Times the observational bound
Oklo (1.8 Gyr ago)	1.14	0.14	\(1.3\times10^{7}\)×
Quasars (\(z\sim2\))	3.0	2.0	\(2.0\times10^{5}\)×
CMB (\(z=1100\))	1101	1100	\(2.8\times10^{5}\)×
Nucleosynthesis (\(z=4\times10^8\))	\(4\times10^{8}\)	\(4\times10^{8}\)	\(4\times10^{10}\)×

VSL — where the
surgery went wrong "The speed of light was faster in the past" holds two different things.
Cut them apart and the observable content is, entirely, "\(\alpha\) varies".

01There are four \(c\)s to begin with

02What VSL actually claims is a variation of \(\alpha\)

03How tightly is \(\alpha\) pinned?

04The heart — how much must \(c\) change to solve the horizon problem?

05The VSL that survives — and the price of surviving

06The reveal — three fates

07Not to be confused — the running of \(\alpha\)

Summary — the failure was not moving \(c\)

Measurement	Epoch	Bound on \(\|\Delta\alpha/\alpha\|\)	Bits pinned
Laboratory (CODATA 2022)	today	\(1.6\times10^{-10}\)	32.5 bit
Atomic clocks (13.8 Gyr extrapolation)	\(z\simeq0\)	\(1.4\times10^{-8}\)	26.1 bit
Oklo natural reactor	1.8 Gyr ago	\(1.1\times10^{-8}\)	26.4 bit
Quasar absorption lines	\(z\sim2\)	\(1.0\times10^{-5}\)	16.6 bit
CMB	\(z=1100\)	\(4.0\times10^{-3}\)	8.0 bit
Nucleosynthesis	\(z=4\times10^{8}\)	\(1.0\times10^{-2}\)	6.6 bit

	Does \(\alpha\) move?	Collision with observation	Predictions
Power-law VSL	yes (\(O(1)\))	ten orders over at nucleosynthesis	falsified
Phase-transition VSL	only before nucleosynthesis	none	zero in the observable era
c·t=const (conformal)	exactly invariant	none	notation; zero in itself

VSL — where thesurgery went wrong "The speed of light was faster in the past" holds two different things.Cut them apart and the observable content is, entirely, "\(\alpha\) varies".

01There are four \(c\)s to begin with

02What VSL actually claims is a variation of \(\alpha\)

03How tightly is \(\alpha\) pinned?

04The heart — how much must \(c\) change to solve the horizon problem?

05The VSL that survives — and the price of surviving

06The reveal — three fates

07Not to be confused — the running of \(\alpha\)

Summary — the failure was not moving \(c\)

VSL — where the
surgery went wrong "The speed of light was faster in the past" holds two different things.
Cut them apart and the observable content is, entirely, "\(\alpha\) varies".