Thinking in Waves Episode 4 ／ Part II · Constants as waves
Not a thought experiment. It has already been measured
Differentiating a constant ──
the speed of light became undifferentiable in 1983
"Is the speed of light changing?" has a trap in it. \(c\) is a defined value, so \(dc/dt=0\) is a convention.
Only dimensionless quantities carry meaning ── \(\alpha\), \(\mu=m_p/m_e\), and \(G\).
Their derivatives have actually been measured. And each experiment corresponds to a frequency band.
Last time we framed it as "a physical constant = the \(\omega=0\) component".
So is there anything at \(\omega\ne0\)? That is an experimental question, and the answer is already in.
01The first trap ── \(c\) cannot be measured
The metre was defined as the distance light travels in \(1/299792458\) s
Measuring \(c\) is no longer possible in principle. The 2019 SI revision extended this treatment to seven constants.
| Constant | Name | Defined value | Since |
|---|---|---|---|
| \(\Delta\nu_{\mathrm{Cs}}\) | caesium hyperfine transition | 9 192 631 770 Hz | 1967 |
| \(c\) | speed of light | 299 792 458 m/s | 1983 |
| \(h\) | Planck constant | \(6.62607015\times10^{-34}\) J·s | 2019 |
| \(e\) | elementary charge | \(1.602176634\times10^{-19}\) C | 2019 |
| \(k_B\) | Boltzmann constant | \(1.380649\times10^{-23}\) J/K | 2019 |
| \(N_A\) | Avogadro constant | \(6.02214076\times10^{23}\) /mol | 2019 |
| \(K_{\mathrm{cd}}\) | luminous efficacy | 683 lm/W | 1979 |
The heart of §01
These seven differentiate to exactly zero. Because they are definitions.
One cannot even ask whether they are changing.
What is left, then ── dimensionless quantities.
"A universe where \(c\) doubled" is the same universe with a redefined metre.
But "a universe where \(\alpha\) doubled" has different chemistry ── a genuinely different universe.
Relation to the sister series
"Dimensionful quantities are bookkeeping; only dimensionless ones are physics" ── this is the slogan established in Part VI of "Understanding c·t = constant". This episode simply restates it from the wave side; it is not a new claim.
What is this series' own contribution starts here ── translating each experiment into a frequency band.
02The derivatives that have been measured
| Method | Target | Timescale probed | Bound \(|\Delta x/x|\) | Note |
|---|---|---|---|---|
| Optical clock comparison | \(\alpha\) | 10 yr | \(10^{-17}\) | Al⁺/Hg⁺ and others |
| Molecular clocks | \(\mu\) | 10 yr | \(10^{-16}\) | rovibrational transitions |
| Oklo natural reactor | \(\alpha\) | 1.8 Gyr | \(10^{-8}\) | from the Sm-149 resonance position |
| Meteorites (Re-187 decay) | \(\alpha\) | 4.6 Gyr | \(10^{-7}\) | half-life depends strongly on \(\alpha\) |
| Quasar absorption lines | \(\alpha\) | 10 Gyr | \(10^{-5}\) | claims of variation exist (disputed) |
| CMB | \(\alpha\) | 13.4 Gyr | \(10^{-2}\) | recombination, \(z\approx1100\) |
| Big Bang nucleosynthesis | \(\alpha\) | 13.8 Gyr | \(10^{-2}\) | \(z\approx10^9\); the oldest probe |
| Lunar laser ranging | \(G\) | 50 yr | \(5\times10^{-12}\) | \(|\dot G/G|<10^{-13}\)/yr |
★ \(\alpha\) has been constant to seven decimal places over the whole history of the universe
03Translating this into frequency bands ── the main point
Suppose a constant oscillates as \(x(t)=x_0(1+A\sin\omega t)\). Each experiment then gives a frequency-dependent bound on \(A\).
| Frequency \(f\) [Hz] | Period | Bound on \(\alpha\) amplitude | Dominant experiment |
|---|---|---|---|
| \(10^{-9}\) | 32 yr | \(5.0\times10^{-17}\) | optical clocks |
| \(10^{-12}\) | \(3.2\times10^4\) yr | \(5.0\times10^{-14}\) | optical clocks |
| \(10^{-15}\) | \(3.2\times10^7\) yr | \(5.0\times10^{-11}\) | optical clocks |
| \(10^{-17}\) | 3.2 Gyr | \(5.0\times10^{-9}\) | optical clocks (up to here) |
| \(10^{-18}\) | 32 Gyr | \(2.8\times10^{-8}\) | ★ Oklo takes over |
| \(10^{-21}\) | \(3.2\times10^4\) Gyr | \(2.8\times10^{-5}\) | Oklo |
| \(10^{-24}\) | \(3.2\times10^7\) Gyr | \(2.8\times10^{-2}\) | Oklo |
Conclusion of §03
The lead changes hands at a period of about 32 Gyr (twice the age of the universe). Faster oscillations belong to optical clocks, slower ones to the Oklo natural reactor.
In baseline, cosmological probes win by 8 orders (10 yr vs 1.8 Gyr). In precision, optical clocks win by 9. On balance, a tabletop clock fights the history of the universe to a draw.
★ And as \(f\to0\), every experiment's bound diverges. That is not a failure but a definition ── as we saw last time, the \(\omega=0\) component is the value of the constant.
04But ── the framework itself has a hole
Everything above assumed a stationary sinusoid. Cosmic history is not stationary.
| Form of change | Content | Which experiment sees it |
|---|---|---|
| A step | one jump at some epoch, constant thereafter | ★ invisible to clocks. Only CMB and nucleosynthesis |
| Damped oscillation | large swings in the early universe, quiet now | same; scalar-field relaxation models have this shape |
| Monotonic drift | slow change proportional to cosmic age | visible to clocks, but extrapolation is unreliable |
The heart of §04 ── the first limit of "see everything as waves"
Fourier is the tool for things that oscillate the same way forever.
Write a one-off event in frequency and it spreads thinly over all frequencies, failing to trip any single band's bound.
⇒ That is why cosmological observations cannot be discarded even when clocks beat them on sensitivity.
This limitation is worth remembering throughout the series ── the language of waves quietly assumes stationarity.
05And on the high-frequency side?
So far, slow change. Fast oscillations are searched for too.
The field's mass \(m\) sets the oscillation frequency
| Field mass [eV] | Frequency [Hz] | Period | Instrument that searches |
|---|---|---|---|
| \(10^{-22}\) | \(2.4\times10^{-8}\) | 1.3 yr | annual modulation in atomic clocks |
| \(10^{-18}\) | \(2.4\times10^{-4}\) | 1.1 hr | atomic clocks |
| \(10^{-15}\) | \(2.4\times10^{-1}\) | 4.1 s | optical cavities |
| \(10^{-12}\) | \(2.4\times10^{2}\) | 4.1 ms | ★ a by-product of interferometers (LIGO etc.) |
"What comes out if you differentiate a constant" has in practice become the dark-matter search itself. So far no significant signal in any band.
Exercises
- How should a claim that "the speed of light has changed" be tested?
Show answer
It cannot be ── not as stated. \(c\) is a defined value, so there is nothing to change.
To make it meaningful one must ask "has \(\alpha=e^2/4\pi\varepsilon_0\hbar c\) changed?" or "has \(m_p/m_e\) changed?". And the moment you translate into a dimensionless quantity, you can no longer tell whether \(c\), \(e\) or \(\hbar\) did the changing.
This is why varying-speed-of-light (VSL) theories draw the criticism they do. - Why does the bound weaken at low frequency?
Show answer
Because a clock measures a rate. An oscillation of amplitude \(A\) and angular frequency \(\omega\) has maximum rate \(A\omega\).
Given a rate bound \(R\), the amplitude bound is \(Athe smaller \(\omega\), the larger an amplitude is allowed at the same rate.
Two-epoch comparisons lose sensitivity the same way when \(\omega\Delta t\ll1\): not even half a cycle has elapsed. - Using Episode 1's language, why is a step change hard to catch with Fourier?
Show answer
The spectrum of a step goes as \(1/\omega\) and spreads thinly over all frequencies.
At any single frequency the amplitude is small and trips no bound — yet summed over all bands there is a finite jump.
This is a direct consequence of the wave language assuming stationarity ── switch to a time-localised basis such as wavelets and you can catch it, but then the notion of "frequency" itself blurs.
Summary Constants you can differentiate, and constants you cannot
\(c\) has been a defined value since 1983. \(dc/dt=0\) is a convention, not a measurement. The 2019 SI revision gave the same treatment to \(h, e, k_B, N_A\) and others, seven in all. Only dimensionless quantities carry meaning ── \(\alpha\), \(\mu=m_p/m_e\), \(G\).
Their derivatives have been measured ── \(|\dot\alpha/\alpha|<10^{-17}\)/yr, \(|\dot\mu/\mu|<10^{-16}\)/yr, \(|\dot G/G|<10^{-13}\)/yr. Over the whole history of the universe \(\alpha\) has been constant to seven decimal places.
Translated into frequency bands, the lead changes hands at a period of 32 Gyr ── faster oscillations to optical clocks, slower ones to Oklo. And as \(f\to0\) the bound diverges. Naturally: the \(\omega=0\) component is the value of the constant.
But the framework has a hole. Fourier assumes stationarity, so a history like "one jump and then flat" spreads thinly over all frequencies and escapes. That is why cosmological observations cannot be discarded even when they lose on sensitivity.
"Differentiating the constants" is already an experimental programme.
The answer, so far, is zero in every band.
Next time we differentiate the same \(\alpha\) with respect to energy. The answer is not zero.
\(1/\alpha\) moves from 137.036 to 128.940. And that change is, literally, an integral.
This document is Episode 4 of the series "Thinking in Waves". The computation is in kenshou/teisu.js. On the numbers in this episode: every measurement in §02 is a summary of published values, not measured here, and rounded to show orders of magnitude (for instance the optical-clock bound \(|\dot\alpha/\alpha|<10^{-17}\)/yr stands for results such as Rosenband et al. 2008, \((-1.6\pm2.3)\times10^{-17}\)/yr). The quasar-absorption claim of \(\Delta\alpha/\alpha\approx10^{-5}\) is unresolved, with an ongoing dispute over systematics ── this episode does not treat it as fact. The frequency dependence of the amplitude bounds in §03 is our own estimate assuming a sinusoid, and differs from the published analyses of the individual experiments (which are done model by model). The mass-to-frequency correspondence for ultralight scalars in §05 uses only the simple relation \(f=mc^2/h\) and says nothing about coupling strength. "Dimensionful is bookkeeping, only dimensionless is physics" is the framing established in the sister series "Understanding c·t = constant". ── To print, use your browser's Print → Save as PDF.