Thinking in Waves Episode 20 ／ Part VIII · Differentiating the effective constants

What appears when time is measured on a logarithmic scale

The $c\cdot t=$const coordinate system ──
the wave equation became the critical $1/r^2$ Define $c(t)=R/t$ and the speed of light is differentiable ($\dot c/c=-1/t=-H$).
Write the wave equation in that system and ── $\ddot\psi+(A^2/t^2)\psi=0$.
★ Episode 13's "fall to the centre" equation came out verbatim.

Prerequisites: Episodes 4, 7, 13Computation: kenshou/cttei.js

"$c$ is a defined value and cannot be differentiated" has been this series' premise since Episode 4.
But hold $c\cdot t$ fixed and $c$ becomes a function of $t$. We tried it, and ended up somewhere unexpected.

01Define the system and differentiate the speed of light

Just a definition $$c(t)\cdot t=R=\text{const}\quad\Longrightarrow\quad c(t)=\frac{R}{t}\quad\Longrightarrow\quad \frac{dc}{dt}=-\frac{c}{t},\qquad \frac{\dot c}{c}=-\frac1t=-H$$

Quantity	Value
$R=c\cdot t$	$1.304\times10^{26}$ m
present $t$	$4.351\times10^{17}$ s (13.79 Gyr)
$dc/dt$	$-6.89\times10^{-10}$ m/s²
$\dot c/c$	$-2.30\times10^{-18}$ /s $=-7.25\times10^{-11}$ /yr

A defined value has just been differentiated. So is this physics, or a change of units?

02Score it first ── as physics, it is already dead

Episode 7's slogan: only dimensionless quantities mean anything $$\alpha=\frac{e^2}{4\pi\varepsilon_0\hbar c}\quad\Longrightarrow\quad \frac{\dot\alpha}{\alpha}=-\frac{\dot c}{c}=+\frac1t=7.25\times10^{-11}\ \text{/yr}$$ $$\text{Episode 4's bound}:\ \Bigl|\frac{\dot\alpha}{\alpha}\Bigr|<10^{-17}\ \text{/yr}\qquad\Longrightarrow\qquad \textbf{excluded by a factor of }\textbf{7.3}\times\textbf{10}^{\textbf{6}}$$

Conclusion of §02

Read "$c\propto1/t$" as a physical change and it is excluded with seven million to spare.
It survives only if every other constant changes in concert to hold $\alpha$ fixed ── which is to say, a change of units.

★ So far this repeats Episodes 4 and 7. The interesting part starts here ── what a wave looks like in this system.

03The phase of a wave goes as $\ln t$

Just integrate the frequency (Episodes 1 and 3) $$\omega(t)=c(t)k=\frac{Rk}{t}\quad\Longrightarrow\quad \varphi(t)=\int\omega\,dt=Rk\,\ln\frac{t}{t_0}$$

The heart of §03 ── it joins up with Episode 13

★ The phase is proportional to $\ln t$.

In Episode 13 the imaginary-order derivative's phase went as $\ln\omega$. Here it goes as $\ln t$ ── the same form (a Mellin kernel).

The instantaneous frequency goes as $1/t$, so this is a hyperbolic chirp.
★ Episode 13 said a click put through an imaginary-order derivative becomes a hyperbolic chirp ── in a $c\propto1/t$ universe every wave is one from the start.

★ So the natural time variable is not $t$ but $u=\ln t$.
And $d/du=t\,d/dt$ is the generator of scale transformations.

04★ The wave equation becomes Episode 13's critical $1/r^2$

Just write the mode equation $$\frac{d^2\psi}{dt^2}+c(t)^2k^2\psi=0\quad\Longrightarrow\quad \ddot\psi+\frac{A^2}{t^2}\psi=0,\qquad A=Rk=k\cdot(ct)$$ $$\psi=t^s\ \Longrightarrow\ s(s-1)+A^2=0\ \Longrightarrow\ s=\frac12\pm\sqrt{\frac14-A^2}$$

★ Compare with Episode 13's formula

Episode 13 (attractive $1/r^2$ potential): $\ \ s=\dfrac12\pm\sqrt{\dfrac14-\lambda}$
This episode (a wave with $c\cdot t=$const): $\qquad s=\dfrac12\pm\sqrt{\dfrac14-A^2}$

Exactly the same equation. The coupling $\lambda$ there is $A^2=(k\cdot ct)^2$ here ── the square of "how many wavelengths fit in the horizon".

Rewriting as $\psi=\sqrt t\,f(\ln t)$ makes it plainer still.

In logarithmic time it is a plain oscillator $$f''(x)+\Bigl(A^2-\frac14\Bigr)f(x)=0,\qquad x=\ln t,\qquad s_0=\sqrt{A^2-\tfrac14}$$

$A=k(ct)$	$\sqrt{1/4-A^2}$	$s_0=\sqrt{A^2-1/4}$	Behaviour
0	0.5000	──	power law (frozen)
0.30	0.4000	──	power law (frozen)
0.50	0	──	★ exactly critical
0.70	imaginary	0.4899	★ oscillates in $\ln t$
2.00	imaginary	1.9365	oscillates in $\ln t$
20.0	imaginary	19.9937	oscillates in $\ln t$

The heart of §04 ── the critical point is horizon crossing

Criticality sits at $A=1/2$, i.e. $k\cdot(ct)=1/2$ ── a comparison of the wavelength with the horizon size $c\cdot t$.

$A>1/2$ (inside the horizon): it oscillates
$A<1/2$ (outside): it stops oscillating and becomes a power law ── it freezes

As $A\to0$ the exponents are $s=1$ and $s=0$ ── ★ a growing mode and a constant mode. Exactly the standard cosmological result that curvature perturbations are constant outside the horizon.

★★ Episode 13's criticality at $\lambda=1/4$ has become
"inside or outside the horizon".

05Checking numerically

Solving $\ddot\psi+(A^2/t^2)\psi=0$ directly by Runge–Kutta and measuring whether the zero crossings are spaced by $\Delta(\ln t)=\pi/s_0$.

$A$	$s_0$	Theory $\pi/s_0$	Measured spacing	Relative error
0.55	0.229129	13.711034	13.711034	$1.2\times10^{-12}$
1.00	0.866025	3.627599	3.627599	$3.3\times10^{-13}$
2.00	1.936492	1.622311	1.622311	$2.2\times10^{-15}$
10.0	9.987492	0.314553	0.314553	$2.8\times10^{-14}$
30.0	29.995833	0.104734	0.104734	$1.1\times10^{-15}$

Agreement to 12–16 digits. And below criticality ($A=0.2$, $A=0.4$) there is not a single zero crossing over six decades of time ── genuinely frozen.

06Which wavelength is critical in today's universe?

Turn the criticality into a wavelength $$\frac{2\pi(ct)}{\lambda}=\frac12\quad\Longrightarrow\quad \lambda_{\rm crit}=4\pi(ct)=1.64\times10^{27}\ \text{m}=173\ \text{Gly}$$

Wave	Wavelength [m]	$A=k(ct)$	State
Visible light (500 nm)	$5\times10^{-7}$	$1.6\times10^{33}$	oscillating (inside)
Size of a galaxy	$9.5\times10^{20}$	$8.6\times10^{5}$	oscillating
CMB anisotropy (1°)	$2.3\times10^{24}$	360	oscillating
Horizon size	$1.3\times10^{26}$	6.28	oscillating
★ critical wavelength	$1.6\times10^{27}$	0.500	★ exactly critical
Ten times that	$1.6\times10^{28}$	0.050	frozen (outside)

§06 ── it matters when the wavelength equals the horizon

Every observable wave has $A\gg1/2$, far inside (visible light is $A=10^{33}$). There $s_0\approx A$ and the log-periodic effect is invisible.

★ It matters at $A\sim1$ ── which is what happens at horizon crossing during inflation.
The pattern in the cosmic microwave background is the record of waves that passed through this critical point.

07What is actually invariant?

Quantity	Verdict	Content
$c(t)$	× coordinate-dependent	$R/t$; set by the choice of units
$t$	× coordinate-dependent	likewise
$R=c\cdot t$	× coordinate-dependent	what we defined
$A=k(ct)$	◎ invariant	★ how many wavelengths fit the horizon. Dimensionless
$s_0=\sqrt{A^2-1/4}$	◎ invariant	★ angular frequency in logarithmic time. Dimensionless
$\Delta(\ln t)=\pi/s_0$	◎ invariant	zero-crossing spacing. Dimensionless

§07 ── the conclusion of this episode

Choosing the $c\cdot t=$const coordinate system
is the same as measuring time on a logarithmic scale.

Use $u=\ln t$ as time and $c$ drops out of the equation, leaving $f''+(A^2-1/4)f=0$ ── a plain oscillator.
★ And the calculus there, $d/du=t\,d/dt$, is Episode 13's imaginary-order world.

The surviving observables are $A$ and $s_0$ ── both dimensionless. Episode 7's conclusion, word for word, again.

08One more line on the map of axes

Axis / operation	Character	Content	Where
Real frequency $\omega$	oscillation	a full period must pass	Eps. 3, 4
Imaginary frequency $i\lambda$	decay	counting is instant	Ep. 19
Real order $\alpha$	tilts the spectrum	$6\alpha$ dB/oct	Eps. 1, 2
Imaginary order $i\beta$	rotates phase as $\ln\omega$	scale transformation	Ep. 13
★ Logarithmic time $\ln t$	rotates phase as $\ln t$	the $c\cdot t=$const system	★ this episode

§08 ── different ways of saying one thing

Episode 13 said real orders are the tool of time translation and imaginary orders the tool of scale transformation. What this episode adds ──

A $c\cdot t=$const universe is a universe whose basic symmetry
is scale transformation rather than time translation.

Hence the natural tool is the imaginary-order derivative, the natural time is $\ln t$, and the natural conserved quantity is $s_0$ ── ★ all different ways of saying one thing.

Exercises

Is "in the $c\cdot t=$const system the speed of light can be differentiated" a counterexample to Episode 4?
Show answer
Not a counterexample ── the clearest illustration of Episode 4's claim.
What was differentiated was not $c$ but the coordinate choice that writes $c$ as $R/t$. And the observables are only the dimensionless $A=k(ct)$ and $s_0$.
★ Whatever you do to $c$ is cancelled somewhere ── and if it were not, $\alpha$ would move, which is excluded by seven million to one.
Explain the freezing of $A<1/2$ modes in Episode 6's language.
Show answer
They dropped below a cutoff frequency.
In Episode 6, below cutoff the wavenumber went imaginary, the wave stopped propagating and decayed exponentially (evanescent).
Here the same thing happens on the time axis: when $s_0$ goes imaginary the solution stops oscillating in $\ln t$ and becomes a power law.
★ The single phenomenon "the square root goes negative" appears in space as the Yukawa potential and in time as horizon freezing.
What happens to energy conservation in this coordinate system?
Show answer
It does not hold ── and that is the point.
Energy conservation follows by Noether's theorem from time-translation symmetry. Here $c(t)=R/t$ breaks it ($t\to t+a$ changes the equation), and what remains is scale symmetry ($t\to\mu t$ leaves $A$ alone).
★ What is conserved is $s_0$ ── instead of energy, "the frequency in logarithmic time".
★ This is the conserved-quantity version of Episode 13's "real = time translation, imaginary = scale transformation".

Summary　Differentiating the speed of light landed us in logarithmic time

Define $c(t)=R/t$ and the speed of light is differentiable, $\dot c/c=-1/t=-H$. Read as a physical change it is excluded by a factor of seven million against the bound on $\alpha$ ── it survives only as a change of units.

But as a wave something interesting happens. The phase goes as $\ln t$ (every wave is a hyperbolic chirp) and the mode equation becomes $\ddot\psi+(A^2/t^2)\psi=0$.

★ That is Episode 13's critical $1/r^2$ equation, word for word, with $\lambda$ replaced by $A^2=(k\cdot ct)^2$ ── the square of how many wavelengths fit in the horizon.

In logarithmic time $x=\ln t$ it reads $f''+(A^2-1/4)f=0$ ── a plain oscillator (confirmed numerically to 12–16 digits). And the critical point $A=1/2$ is horizon crossing, outside which $s=0$ and $s=1$ give a constant mode and a growing mode ── standard cosmology's "fluctuations freeze outside the horizon".

★ The $c\cdot t=$const coordinate system was time measured on a logarithmic scale.
The calculus there is Episode 13's imaginary-order world.

The surviving observables are $A$ and $s_0$ ── both dimensionless. Episode 7's conclusion, at work again.

Next time Integrate a bit rate [bit/s] and you get an amount of information [bit]. That is calculus too.
★ And Shannon's formula is "bandwidth × time" ── the same shape as Episode 9's uncertainty principle.
Landauer's limit, the Bremermann limit, the holographic bound ── count information as a wave and the limits of physics all join up.

This document is Episode 20 of the series "Thinking in Waves". The computation is in kenshou/cttei.js. On this episode: "$c\cdot t=$const" is a notation treated at length in the sister series "Understanding c·t = constant"; here we revisit it from the wave side. Our $c(t)=R/t$ is a naive model placed on flat spacetime, not a solution of general relativity ── real cosmology uses a scale factor $a(t)$, and in conformal time the structure is similar but an extra term ($a''/a$) appears. So the number "critical at $A=1/2$" is specific to this model and agrees in structure but not in value with the standard horizon-crossing condition. ★ What this episode claims is not a value but an agreement of form ── that the boundary between real and complex exponents corresponds to "inside or outside the horizon". The bound $10^{-17}$/yr on $\dot\alpha/\alpha$ is the published value used in Episode 4. The zero-crossing spacings are our own RK4 computation, agreeing with the analytic solution to 12–16 digits (internal consistency of the model). ── To print, use your browser's Print → Save as PDF.

Quantity	Value
\(R=c\cdot t\)	\(1.304\times10^{26}\) m
present \(t\)	\(4.351\times10^{17}\) s (13.79 Gyr)
\(dc/dt\)	\(-6.89\times10^{-10}\) m/s²
\(\dot c/c\)	\(-2.30\times10^{-18}\) /s \(=-7.25\times10^{-11}\) /yr

01Define the system and differentiate the speed of light

02Score it first ── as physics, it is already dead

03The phase of a wave goes as \(\ln t\)

04★ The wave equation becomes Episode 13's critical \(1/r^2\)

05Checking numerically

06Which wavelength is critical in today's universe?

07What is actually invariant?

08One more line on the map of axes

Summary　Differentiating the speed of light landed us in logarithmic time

Wave	Wavelength [m]	\(A=k(ct)\)	State
Visible light (500 nm)	\(5\times10^{-7}\)	\(1.6\times10^{33}\)	oscillating (inside)
Size of a galaxy	\(9.5\times10^{20}\)	\(8.6\times10^{5}\)	oscillating
CMB anisotropy (1°)	\(2.3\times10^{24}\)	360	oscillating
Horizon size	\(1.3\times10^{26}\)	6.28	oscillating
★ critical wavelength	\(1.6\times10^{27}\)	0.500	★ exactly critical
Ten times that	\(1.6\times10^{28}\)	0.050	frozen (outside)

Quantity	Verdict	Content
\(c(t)\)	× coordinate-dependent	\(R/t\); set by the choice of units
\(t\)	× coordinate-dependent	likewise
\(R=c\cdot t\)	× coordinate-dependent	what we defined
\(A=k(ct)\)	◎ invariant	★ how many wavelengths fit the horizon. Dimensionless
\(s_0=\sqrt{A^2-1/4}\)	◎ invariant	★ angular frequency in logarithmic time. Dimensionless
\(\Delta(\ln t)=\pi/s_0\)	◎ invariant	zero-crossing spacing. Dimensionless

Axis / operation	Character	Content	Where
Real frequency \(\omega\)	oscillation	a full period must pass	Eps. 3, 4
Imaginary frequency \(i\lambda\)	decay	counting is instant	Ep. 19
Real order \(\alpha\)	tilts the spectrum	\(6\alpha\) dB/oct	Eps. 1, 2
Imaginary order \(i\beta\)	rotates phase as \(\ln\omega\)	scale transformation	Ep. 13
★ Logarithmic time \(\ln t\)	rotates phase as \(\ln t\)	the \(c\cdot t=\)const system	★ this episode

01Define the system and differentiate the speed of light

02Score it first ── as physics, it is already dead

03The phase of a wave goes as \(\ln t\)

04★ The wave equation becomes Episode 13's critical \(1/r^2\)

05Checking numerically

06Which wavelength is critical in today's universe?

07What is actually invariant?

08One more line on the map of axes

Summary Differentiating the speed of light landed us in logarithmic time

Summary　Differentiating the speed of light landed us in logarithmic time