c·t = CONST, THAT CLICKS EPISODE 9 ／ Why the blade of 1918 does not reach
Atoms shrink, and the spectral lines do not blur
Substituting
into the atom
Einstein killed Weyl's unified theory with exactly this argument.
Two numbers say why the same blade misses the modern conformal transformation.
In this picture atoms shrink as \(1/t\). The Bohr radius is \(\hbar/(mc\alpha)\), so it gets smaller as the mass grows. Doesn't that blur the spectral lines? This is not idle worry: it is exactly the argument with which Einstein killed Weyl's theory in 1918. This episode says, in two numbers, why that blade cannot reach the modern conformal transformation.
01The blade Einstein used in 1918
A recap of Episode 1 of the previous series. Weyl proposed \(g_{\mu\nu}\to\Omega(x)^2g_{\mu\nu}\), letting the standard of length be re-chosen at every point, and identified the compensating field with the electromagnetic potential. Einstein objected in an appendix to the same paper.
The objection was decisive; Weyl withdrew the theory. One observational fact — that atoms do not remember their history — killed an entire unified theory.
02The modern conformal transformation is doubly protected
Point ① is history, settled in the previous series. Today is ②: even if atoms really are shrinking, why the lines do not blur, in numbers.
03Computing the adiabatic parameter
The adiabatic theorem says: if the Hamiltonian changes slowly enough compared with the level spacing, the system follows its level without jumping out. The test is a ratio of two rates.
Rate of change of the Hamiltonian (here, the growth rate of mass)
$$\frac{\dot m}{m}=\frac{1}{t}=H$$Rate at which a level can respond
$$\frac{\Delta E}{\hbar}$$The ratio — the adiabatic parameter
$$\varepsilon=\frac{\hbar H}{\Delta E}\qquad(\varepsilon\ll1\ \text{means adiabatic})$$Put in today's \(\hbar H_0=1.51\times10^{-33}\) eV.
| Transition | \(\Delta E\) | \(\varepsilon=\hbar H_0/\Delta E\) | Adiabaticity fails at \(t=\hbar/\Delta E\) |
|---|---|---|---|
| Hydrogen Rydberg | 13.6 eV | \(1.1\times10^{-34}\) | \(4.8\times10^{-17}\) s |
| Lyman α | 10.2 eV | \(1.5\times10^{-34}\) | \(6.5\times10^{-17}\) s |
| Visible light | 2 eV | \(7.6\times10^{-34}\) | \(3.3\times10^{-16}\) s |
| Caesium clock (9.19 GHz) | \(3.8\times10^{-5}\) eV | \(4.0\times10^{-29}\) | \(1.7\times10^{-11}\) s |
| 21 cm hyperfine | \(5.9\times10^{-6}\) eV | \(2.6\times10^{-28}\) | \(1.1\times10^{-10}\) s |
Conclusion of §03
Even the slackest transition (21 cm) loses adiabaticity before \(10^{-10}\) s.
Atoms only form at recombination (\(1.2\times10^{13}\) s = 380,000 yr).
Perfectly adiabatic throughout the entire era in which atoms exist.
Figure: age of the universe across, adiabatic parameter \(\hbar H/\Delta E\) up. Adiabaticity fails where the curve rises above the vermilion line (\(\varepsilon=1\)). The grey band is "the era in which atoms exist" — however you move the transition energy, the crossing stays far to the left of it.
Push the slider to the far left (\(10^{-10}\) eV, slacker than any transition one can imagine) and the crossing still does not reach the band. Between the formation of atoms and today, adiabaticity never fails once.
04So how much do the lines blur?
Natural width of Lyman α (spontaneous emission \(A=6.27\times10^8\ \mathrm{s^{-1}}\))
$$\frac{\Delta\nu}{\nu}=\frac{A/2\pi}{\nu}=\frac{9.97\times10^{7}}{2.47\times10^{15}}=4.04\times10^{-8}$$Fractional change in frequency during the emission (\(1/A\) seconds) in this picture
$$\frac{\dot\nu}{\nu}\cdot\frac{1}{A}=\frac{H_0}{A}=\frac{2.30\times10^{-18}}{6.27\times10^{8}}=3.67\times10^{-27}$$The thing this episode most wants to say
The blurring from shrinking atoms is \(9\times10^{-20}\) times the natural width.
── Nineteen orders below. Einstein's blade does not reach.
05Atomic clock limits are satisfied exactly, by zero
| Quantity | Observational bound | This picture predicts |
|---|---|---|
| \(\dot\alpha/\alpha\) | \(1.0(1.1)\times10^{-18}\)/yr (Lange 2021) | exactly 0 |
| \(\dot\mu/\mu\) (\(\mu=m_p/m_e\)) | \(\sim10^{-17}\)/yr (molecular clocks) | exactly 0 |
| Every mass ratio and charge ratio | various | exactly 0 |
The reason is the usual one — numerator and denominator carry the same weight. In \(\alpha=e^2/4\pi\varepsilon_0\hbar c\), all of \(e,\hbar,c\) have weight 0. In \(\mu=m_p/m_e\), both are weight \(-1\). They cancel; there is no way for them to move.
06The reveal — there is no comparison partner
As with \(G\) in Episode 7, read naively that is an outrageous rate. So why is it not measurable? Because when you look for something to compare against, everything has the same weight.
| Compare the size of an atom against… | Its weight | The ratio |
|---|---|---|
| another atom | \(+1\) | invariant |
| a wavelength of light | \(+1\) | invariant |
| a ruler (made of atoms) | \(+1\) | invariant |
| \(c\times\) a clock tick | \(+1\) | invariant |
| nothing at all | ── | there is no claim |
Do to the atom what Episode 3 did to the title of the series and you get this: "atoms are shrinking" is not a sentence until you name what they are shrinking relative to. And when you look, there is no ruler outside atoms that is independent of atoms. The starting point of Episode 2 of the previous series — every ruler bottoms out in atoms — closes here.
① Writing the adiabatic parameter as \(\hbar H/\Delta E\) is the crudest estimate. The exact condition is \(|\langle m|\partial_t H|n\rangle|/(E_n-E_m)^2\ll1\), with matrix elements. This document evaluates only "rate of change of the Hamiltonian ÷ level spacing" by dimensional analysis — but matrix elements are dimensionless \(O(1)\) numbers, so the conclusion at 34 orders does not move.
② The "blurring" is estimated as the frequency change during the radiative lifetime \(1/A\). It is not a full line-shape calculation (natural, Doppler and pressure widths). Read it as an order-of-magnitude argument; laboratory lines are buried under Doppler and collisional widths far larger than this effect.
③ "Satisfies the atomic clock limits exactly by zero" holds when \(c\cdot t=\text{const}\) is implemented as a conformal transformation. In a VSL-type implementation that moves only \(c\), \(\alpha\) moves and collides with these bounds (Extra 3 of the previous series). The same words "the speed of light varies" live or die on the implementation.
④ Weyl's 1918 theory and this operation are different things. Weyl proposed a geometry with a non-integrable length connection (Weyl geometry). Here \(\Omega\) is single-valued and the connection integrable, so there is no second clock effect from the start. The objection was not dodged; there is nothing for it to hit — exactly the account in Episode 1 of the previous series. Non-integrable Weyl geometry, incidentally, remains a live research topic in gravity.
⑤ Recombination is taken as "when atoms form". In fact hydrogen forms in bulk at \(z\simeq1100\), and short-lived bound states existed earlier. The band in the figure is an indication.
Exercises (solvable with this episode's formulas alone)
- State in one line the argument with which Einstein killed Weyl's theory in 1918.
Show the answer
If carrying the standard of length requires a path, the same atom emits light of different wavelength depending on the route it took (the second clock effect). But real spectral lines are sharp — so it is refuted immediately. - Give two reasons the modern conformal transformation is not cut by the same argument.
Show the answer
① \(\Omega(x)\) is single-valued, so the change of length is path-independent (integrable) and the second clock effect cannot occur. ② Even with time-varying mass, the adiabatic parameter \(\hbar H/\Delta E\sim10^{-34}\) induces no transitions. Doubly protected. - When does adiabaticity fail for the 21 cm line (\(\Delta E=5.9\times10^{-6}\) eV)?
Show the answer
\(t=\hbar/\Delta E=6.58\times10^{-16}/5.9\times10^{-6}=1.1\times10^{-10}\) s — 23 orders of magnitude before atoms form at recombination (\(1.2\times10^{13}\) s). It never fails in an era with atoms. - The Bohr radius shrinks at \(7.2\times10^{-11}\)/yr. Why is that not measurable?
Show the answer
Because everything to compare against has weight \(+1\) — other atoms, wavelengths of light, rulers, \(c\times\)clock ticks all shrink by the same factor. There is no ruler outside atoms that is independent of atoms (Episode 2 of the previous series). So "atoms are shrinking" is not a sentence until you name the comparison. - (Harder) Both say "the speed of light varies", yet VSL is killed by atomic clocks and this picture is not. What is the difference?
Show the answer
VSL moves \(c\) alone while fixing \(e,\hbar\), so \(\alpha\) moves and collides with the atomic clock bound (\(\dot\alpha/\alpha<10^{-18}\)/yr). This picture moves the whole set together, so \(\alpha\) is exactly invariant. Life or death turns not on how you move dimensionful quantities but on whether you protect the dimensionless ones (Extras 3 and 4 of the previous series). The same pattern as \(G\) in Episode 7.
Summary — Einstein's blade is stopped twice
Atoms shrink as \(1/t\) here. In 1918 Einstein killed Weyl's unified theory by attacking precisely this kind of variation — if carrying a standard of length needs a path, atoms remember their history and spectral lines blur. A decisive objection.
The same blade cannot reach the modern conformal transformation, for two reasons. ① \(\Omega(x)\) is single-valued, so there is no path dependence at all (Episode 1 of the previous series). ② And the second is today's calculation: the adiabatic parameter \(\varepsilon=\hbar H/\Delta E\) is \(1.1\times10^{-34}\) for the hydrogen Rydberg. Adiabaticity would fail at \(t=\hbar/\Delta E=4.8\times10^{-17}\) s, and even for the slackest 21 cm line at \(1.1\times10^{-10}\) s. Atoms only form at \(1.2\times10^{13}\) s (380,000 yr), so the entire era of atoms is perfectly adiabatic.
We also counted the blurring: the frequency change over a radiative lifetime is \(H_0/A=3.7\times10^{-27}\) — \(9\times10^{-20}\) of Lyman α's natural width \(4.0\times10^{-8}\), nineteen orders below. And the atomic-clock bounds on \(\dot\alpha/\alpha\) and \(\dot\mu/\mu\) are met by predictions of exactly zero (numerator and denominator share a weight).
The reveal took its usual shape. The Bohr radius really does shrink at \(7.2\times10^{-11}\)/yr and still cannot be measured — because everything to compare against has weight \(+1\). Other atoms, wavelengths, rulers, all shrink alike. "Atoms are shrinking" is not a sentence until you name the comparison. Apply Episode 3's surgery to the atom and you arrive right back at Episode 2 of the previous series: every ruler bottoms out in atoms.
This document is Episode 9 of "c·t = const, That Clicks", written for physics-minded high-school and university readers. Weyl's (1918) gauge theory of length, Einstein's objection to it (a non-integrable length connection produces a second clock effect, yet atomic spectra are sharp), and the fact that the modern conformal transformation restricts \(\Omega\) to a single-valued function — the integrable case, with no second clock effect — are established history and physics (Episode 1 of the previous series). The adiabatic theorem, and the fact that adiabaticity is governed by the ratio of the Hamiltonian's rate of change to the level spacing, are standard. The \(\varepsilon=\hbar H/\Delta E\) used here is a crude dimensional estimate; the exact condition involves the matrix element \(\langle m|\partial_tH|n\rangle/(E_n-E_m)^2\) — but those are dimensionless \(O(1)\) numbers, so the order of magnitude of the conclusion is unaffected. The table (\(\hbar H_0=1.51\times10^{-33}\) eV, \(1.1\times10^{-34}\) for the Rydberg, failure time \(t=\hbar/\Delta E\)) and the line-width comparison (\(H_0/A=3.7\times10^{-27}\) against Lyman α's natural width \(4.04\times10^{-8}\), a ratio of \(9\times10^{-20}\)) are computed here. Lyman α's \(A=6.265\times10^8\ \mathrm{s^{-1}}\) and \(\nu=2.466\times10^{15}\) Hz are standard values. The width estimate uses the frequency change over the radiative lifetime and is not a full line-shape calculation (Doppler and collisional widths dominate in the laboratory). The atomic clock bound \(\dot\alpha/\alpha=1.0(1.1)\times10^{-18}\)/yr is Lange et al. (2021, PRL 126, 011102). "Satisfies the bounds exactly by zero" applies when \(c\cdot t=\text{const}\) is implemented as a conformal transformation; a VSL-type implementation moving only \(c\) makes \(\alpha\) vary and collides with these bounds (Extra 3 of the previous series). Non-integrable Weyl geometry remains a research topic today. Linear expansion (\(c\cdot t=\)const, \(R_h=ct\)) is a minority model under examination and conflicts with nucleosynthesis when extrapolated into the early universe (Lewis, Barnes & Kaushik 2016). 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).