Bonuses 3 and 4 were about "protecting α." This time we flip it: once you decide to move α, we count what else you can still hold fixed.
The bonus issues so far have been written from the side that says "α is something to protect." In Bonus 3, VSL failed because it couldn't protect α; in Bonus 4, we saw that α being invariant is the very core of the equivalence condition. This time we turn the viewpoint inside out: if we really decide to move α, how much "freedom to hold things fixed" do we have left? The answer can be counted with surprising precision — and it turns out to be a reprise, one level down, of this series' watchword: "c = constant and c·t = constant are the same thing; they just fix different quantities." This is part one of a trilogy.
Let's write the fine-structure constant in the form from Episode 2.
The ingredients are the four quantities \(c,\ e,\ \varepsilon_0,\ \hbar\). And yet, out of these four, there is only one "number with the units gone" you can build: \(\alpha\). This is no accident — it has a deep meaning. Against the four kinds of units (length, time, mass, charge), these four constants carry only three units' worth of independent information. So the leftover one unit's worth spills over as a "relation with the units gone" — that is \(\alpha\).
From here comes the iron rule of fixing.
Of \(c,e,\varepsilon_0,\hbar\), the number you can freely fix by convention (units, gauge) is at most 3. The remaining one is always dragged along by \(\alpha\) and moves.
So once you've decided to "move \(\alpha\)," this "one that moves" can never be erased.
From the very beginning, this series has spoken in a coordinate frame where \(c=\text{constant}\). In other words, we've already spent one fixing-slot on c. By the iron rule, then, we can additionally fix at most 2 of \(\{e,\varepsilon_0,\hbar\}\). The third always moves, and that "one that moves" is exactly the bookkeeping face of "α changing." Depending on whom you assign to be the mover, the same physics can be written three ways.
| What you fix (= the yardstick) | The mover | What it means |
|---|---|---|
| c, ε₀, ħ | e (∝√α) | A theory where "electric charge changes over time." Bekenstein-type varying-e is this one. |
| c, e, ħ | ε₀ (∝1/α) | A picture where "the vacuum's response (how hard it is for electricity to get through) changes." |
| c, e, ε₀ | ħ (∝1/α) | A picture where "the quantum step size changes." |
Whichever you choose, the colors of atoms and the rhythms of clocks — that is, the observations are completely identical. The only difference is the bookkeeping choice of "whom to make the mover." In the figure below, drag to watch how, when \(\alpha\) grows a little, each of the three movers shifts (and yet \(\alpha\) stays the same).
What I want you to notice here is that what just happened is a strikingly close, scaled-down copy of the structure we saw in Bonuses 3 and 4. Back then it was "fix \(c\), or fix \(c\cdot t\) — same physics, just different things fixed." This time we go one level down: "fix \(e\), or fix \(\hbar\), or fix \(\varepsilon_0\) — same physics, just different things fixed." "Whom you make the mover is gauge (a rebalancing of the ledger); what shows up in observation is only the dimensionless \(\alpha\)" — the same single principle keeps showing its face over and over, at one hierarchy level after another.
Here is the decisive difference from Bonuses 3 and 4, and the point you absolutely must not miss. In Bonus 4's \(c\cdot t=\text{constant}\), \(\alpha\) was invariant. So that was pure gauge (freedom of viewpoint), with zero observational difference — it was "the same physics." But this time, \(\alpha\) itself moves. The moving of \(\alpha\) is a genuine physical change that no choice of units or gauge can erase. It leaves traces in observation: atomic clocks tell us "α has moved by less than \(10^{-17}\) per year," and \(\dot\alpha/\alpha\) is still being hunted in the absorption lines of distant quasars.
"Consider a coordinate frame in which \(\alpha\) changes" should, more precisely, be —
"Add one new piece of physics (= assume α moves), then choose which constant to blame it on."
The coordinate = gauge freedom that remains is only in "which constant to blame it on"; the fact itself that "α moves" cannot be erased in any coordinate frame.
First, you cannot fix all four of \(\{c,e,\varepsilon_0,\hbar\}\). Freezing all of them would force \(\alpha=\text{constant}\), which contradicts the premise "move α." You can fix at most three.
Also, \(\alpha\) is only one of the dimensionless numbers. There's also the proton-to-electron mass ratio \(m_p/m_e\), the strong coupling, gravity's \(\alpha_G=Gm^2/\hbar c\), and more — there are several "numbers with the units gone." This time we assumed "only \(\alpha\) moves, all other ratios fixed," but every extra dimensionless number you let move adds one independent new piece of physics, each carrying its own separate observational constraints. You don't get it for free.
The four constants \(c,e,\varepsilon_0,\hbar\) that build \(\alpha\) are, dimensionally, only "3 people's worth" independent (one constraint = that's α). So by convention you can fix at most three; the remaining one always moves along with α. In this series, where c is already fixed, you can additionally fix at most two of \(\{e,\varepsilon_0,\hbar\}\). Whom you make the mover is gauge (ledger), and varying-e, varying-ε₀, and varying-ħ are all restatements of the same physics — this was a one-level-down reprise of "c = constant vs. c·t = constant."
The decisive difference from Bonus 4, however, is that \(\alpha\) itself moves. That is real physics that gauge can't erase, and it shows up in observation. So the right phrasing is not "a coordinate frame where α moves" but "choose which constant to blame the physics of moving α on." The freedom you can fix is three, but the fact that α moves can't be erased in any coordinate frame — this is the starting point of the trilogy.
Print / PDF: ⌘+P (on Windows, Ctrl+P). On screen, the slider lets you see that the same α can be written in three ledgers. Click "See the answer" to open a solution.