Thinking in Waves Episode 3 ／ Part II · Constants as waves
From one line of code to the physical constants
Is the constant of integration
the physical constant? ── what lives at \(\omega=0\)
Integration is \(\times1/(i\omega)\). It diverges at \(\omega=0\), so the DC component alone is undetermined.
That is what the "constant of integration" really is ── zero frequency, a component of infinite period.
And with only an observation window \(T\), we cannot tell a wave of period \(>T\) from a constant.
The spectral-method code in Episode 1 had exactly one special line.
This time we connect that line to the physical constants.
01The line in question
★ DC cannot be defined
| Operation | What happens at \(\omega=0\) | Meaning |
|---|---|---|
| Differentiation | kills DC | \(\times(i\cdot0)=0\). A constant term vanishes on differentiation ── information is lost |
| Integration | cannot determine DC | \(1/(i\cdot0)\) diverges ── ★ this is the constant of integration |
The heart of §01
The constant of integration was the zero-frequency component.
Differentiation crushes DC, so it throws information away. The inverse operation therefore cannot restore what was thrown away ── and the remainder is written "\(+C\)".
That \(C\) you were told never to forget in school is the origin of the spectrum.
02What is zero frequency?
\(\omega=0\) means infinite period. No real observation can measure that. All we can measure are periods shorter than our window.
A wave with period longer than \(T\) never completes a single cycle ── indistinguishable from a constant
| Observation window | Length | Lowest resolvable frequency | Note |
|---|---|---|---|
| A human lifetime | 80 yr | \(4.0\times10^{-10}\) Hz | |
| History of precision measurement | 400 yr | \(7.9\times10^{-11}\) Hz | since Newton |
| Age of the Earth | 4.6 Gyr | \(6.9\times10^{-18}\) Hz | Oklo and meteorites usable |
| Age of the universe | 13.8 Gyr | \(2.3\times10^{-18}\) Hz | ★ the limit in principle |
Conclusion of §02 ── rewriting the definition of a constant
A physical constant ＝ a component whose period exceeds the age of the universe
It is not that it "does not change" but that we have no means of observing a change.
So to the question "are constants integrals?" one can answer: yes — but the interval of integration is the history of the universe, and anything slower than that is invisible in principle.
03Some things this view explains for free
③ sets up the next episode
When we ask "what happens if we differentiate a constant", differentiating a quantity that is merely a convention about units gives nothing but zero.
And the speed of light \(c\) is exactly such a quantity ── it has been a defined value since 1983.
That is where we start next time.
04When a constant of integration becomes physics
| Setting | Constant of integration | What it really is |
|---|---|---|
| \(\int F\,dx\) | \(V_0\) | the zero of potential. Does not appear in observables (only differences matter) |
| Einstein's equations | \(\Lambda\) | ★ the cosmological constant. It does appear in observables |
| Third law of thermodynamics | \(S_0\) | the zero of entropy. Nernst fixed it at 0 |
| Electromagnetic potential | gauge | a degree of freedom absent from physics — hence gauge symmetry |
The heart of §04 ── there are two kinds
Constants of integration come in two kinds: those that show up in observations, and those that do not.
The ones that do not (\(V_0\), gauge) become symmetries ── because changing them does nothing.
The ones that do (\(\Lambda\)) become new constants ── because they must be measured.
★ The cosmological constant \(\Lambda\) is literally a case of "a constant of integration becoming physics". Einstein noticed this freedom when writing the equations, discarded it once, and observation later brought it back.
05So can \(\omega=0\) be measured at all?
No. And it is not a matter of technology but of definition.
Constraining it means saying "\(\alpha\) is not 1/137",
which is a statement about measurement error, not about detecting change
Conclusion of this episode
\(\omega=0\) is not a thing to be measured; it is the value itself.
So the question "is a constant changing?" must always be translated into a search for components at \(\omega\ne0\).
And the smaller \(\omega\), the worse the experimental sensitivity ── next time we measure that "sensitivity spectrum".
Exercises
- In what sense does differentiation throw information away?
Show answer
\(\times(i\omega)\) multiplies by zero at \(\omega=0\), so the DC component vanishes completely.
Just as multiplying by zero cannot be undone, this is an irreversible operation. Hence the inverse (integration) always leaves an undetermined constant. - Why does the zero of potential \(V_0\) not appear in physics while the cosmological constant \(\Lambda\) does?
Show answer
Because gravity couples to energy itself.
In Newtonian mechanics the force is \(-dV/dx\), so \(V_0\) cancels; in general relativity the vacuum energy density curves spacetime. The symmetry "nothing happens if you shift the origin" fails for gravity.
This is also the doorway to the cosmological constant problem (calculated and observed vacuum energy differ by \(10^{120}\)). - Give an everyday example of "a period longer than the window looks constant".
Show answer
Continental drift. Over a human lifetime (resolving down to \(4\times10^{-10}\) Hz) the continents' positions look constant.
Through the geological window (\(10^{-15}\) Hz) they are obviously moving. Whether something is "constant" is not a property of the object but of its relation to the observation window.
There is no argument that the physical constants must be exempt from the same logic.
Summary What lives at \(\omega=0\)
Integration is \(\times1/(i\omega)\), which diverges at \(\omega=0\) and cannot fix the DC component. That is what the constant of integration "\(+C\)" really is ── the origin of the spectrum.
And \(\omega=0\) means infinite period. With only a window \(T\), we cannot distinguish a wave of period \(>T\) from a constant. Even using the age of the universe, \(2.3\times10^{-18}\) Hz is the floor.
Several things then connect for free ── conserved quantities are DC components; the boundary between law and initial condition is the boundary between \(\omega\ne0\) and \(\omega=0\); the choice of units lives at DC too.
Constants of integration come in two kinds. Those invisible to observation (the zero of potential, gauge) become symmetries; those visible (the cosmological constant \(\Lambda\)) become new constants. \(\Lambda\) is literally a constant of integration that became physics.
A physical constant ＝ a component whose period exceeds the age of the universe.
Not unchanging — merely beyond any means of observing change.
This is not a thought experiment ── it has already been measured. Optical clocks, the Oklo natural reactor, quasar absorption lines, lunar laser ranging.
And we stumble immediately ── differentiating the speed of light gives zero, but that is a convention, not a measurement.
This document is Episode 3 of the series "Thinking in Waves". The computations are in kenshou/nami.js and teisu.js. On this episode: the formulation "a physical constant = a component whose period exceeds the age of the universe" is our own framing, not a standard definition (the standard phrasing is "an input parameter of the theory"). The Fourier resolution \(\Delta f=1/T\) is exact, however, and so is the consequence that variation with period \(>T\) is indistinguishable from a constant. The classification of constants of integration (observable / not) is also our framing, not standard terminology. There are several positions on regarding \(\Lambda\) as a constant of integration of Einstein's equations; here we use it in the sense of "the lowest-dimension term one may add to the action". There is no new computation in this episode; it bridges the measurements of Episode 1 to Episode 4. ── To print, use your browser's Print → Save as PDF.