c·t = CONST, THAT CLICKS EPISODE 16 ／ Part II wrap-up
Across nine fields, not one exception
Only one thing
ever moves
From gravity to biology, the same notation, counting what moves.
Everything that moved was dimensionful; everything that did not was dimensionless.
Part II's ten episodes fold into one page. Gravity, quantum mechanics, atoms, heat and information, light, the vacuum, fluids, phase transitions, chemistry and biology — wherever we substituted, only one thing ever moved. And listing everything that did not move gives, directly, an inventory of what physics is. Finally we draw the line between where this notation genuinely helps and where it is completely powerless, and close Part II.
01Ten episodes, one line each
| Ep. | Substituted into | What moved | What did not |
|---|---|---|---|
| 7 | gravity | \(G\) (\(\div a^2\)), \(g\), \(r_s\), \(T_H\) | \(\alpha_G\), Dirac's large numbers, BH entropy, strain \(h\) |
| 8 | quantum mechanics | de Broglie wavelength, levels, Compton wavelength | the equation's form, \(\Delta x\Delta p\), \(S/\hbar\), tunnelling |
| 9 | the atom | Bohr radius (\(7.2\times10^{-11}\)/yr) | spectral lines, \(\alpha\), \(\mu=m_p/m_e\) |
| 10 | heat and information | temperature, Landauer cost | entropy, Boltzmann factor, the second law |
| 11 | light | (nothing moves) | \(n_\gamma\), \(\rho_\gamma\), \(T\), \(\lambda\), \(s/n\), \(\eta_b\) |
| 12 | the vacuum | \(\rho_\Lambda\) (\(\propto t^4\)), \(\rho_m\), the ranking | \(\rho_\Lambda/M_{\rm Pl}^4\), \(w=-1\), equality epochs |
| 13 | fluids and turbulence | \(\rho\), \(\eta\), \(L\) (all differently) | Re, Ma, Pr, Fr, We, St, the \(-5/3\) law |
| 14 | phase transitions | ── | critical exponents, \(\Delta\) (but the weight table itself gets error bars) |
| 15 | chemistry and biology | mass, length, metabolic rate, lifespan | Arrhenius factor, lifetime heartbeats, genetic information |
Conclusion of §01
The left column is entirely dimensionful; the right column entirely dimensionless.
Across nine different fields, not one exception.
02A map of the weights
| Weight | In this picture | What lives there |
|---|---|---|
| \(+3\) | \(\times a^3\) | number density |
| \(+2\) | \(\times a^2\) | area, \(1/G\) |
| \(+1\) | \(\times a\) | length, time, wavelength, Bohr radius, Compton wavelength, lifespan, kinematic viscosity, Kolmogorov length |
| \(0\) | invariant | velocity, \(c\), \(\hbar\), \(e\), \(\alpha\), \(\alpha_G\), entropy, bits, phase, every ratio, every exponent |
| \(-1\) | \(\div a\) | mass, energy, temperature, frequency, gravitational acceleration, Lyapunov exponent, dissipation rate |
| \(-2\) | \(\div a^2\) | curvature \(R\), metabolic rate, tidal force |
| \(-3\) | \(\div a^3\) | viscosity, surface tension |
| \(-4\) | \(\div a^4\) | energy density, mass density, pressure |
(The "in this picture" column is written in the direction \(\tilde X=a^{-w}X\). A length of weight \(+1\) shrinks as \(\div a\); a mass of weight \(-1\) grows as \(\times a\) — sign and direction are opposite, which is always the confusing part.)
Figure: the map of weights. Drag back in time and only the weight-0 column stays still; the others scatter as \(a^{-w}\) — and only the still column is observable.
03Where it helps, and where it is powerless
The thing this episode most wants to say
A conformal transformation is a tool that touches only "size".
So it is powerful where size is the protagonist and powerless where shape is.
And that boundary is not a defect of the tool but its very definition.
04Except that the weight table itself gets error bars
The proviso found in Episode 14 belongs here. §02's table is a classical approximation built by dimensional analysis. In field theory \(\Delta=\Delta_{\text{classical}}+\gamma\), and \(\gamma\) is an object of measurement — \(\gamma_\sigma=0.0181489(10)\) in the 3D Ising model, seven digits.
| Layer | Does it break? | Why |
|---|---|---|
| Measured dimensionless quantities (ratios, exponents, bits) | no | they are observables |
| The values of the weights themselves | yes (quantum corrections) | Episode 14, \(\gamma_\sigma\) |
| Geometric weights (length, time) | no | they define the metric |
So precisely — not "safe because dimensionless" but "safe because observable". Part V digs into the places where this distinction bites (anomalies, ghosts, rotating spacetimes).
05The map so far, with Part I
| Part | What it did | Conclusion |
|---|---|---|
| I | build the notation (Episodes 1–6) | \(c\cdot t=\)const is notation, not a model |
| II | substitute it into every equation in reach (Episodes 7–16) | everywhere, only one thing moves. It touches only size |
| III | measure as information (Episodes 17–26) | to come |
① Part II derived no new physics whatsoever. All it did was rewrite known laws in a different notation and count what moves and what does not. The reason it still seems worth doing is that the inventory "what does not move = physics" was confirmed without exception across nine fields.
② The "moved / did not move" classification is for rewriting the whole universe at once. Fixed laboratory conditions (a liquid of given viscosity, a thermostat, an external field) do not transform by themselves — the caveat repeated in Episodes 8 ①, 13 ① and 15 ③.
③ The sign convention for weights is a convention. This series takes \(\tilde X=\Omega^{w}X\) (\(\Omega=1/a\)), with lengths at \(w=+1\) and masses at \(-1\). Other sources use the opposite sign, or write \(\Delta=-w\) — always check the convention before comparing.
④ The "helps / powerless" line is this series' own assessment. There are fields — conformal field theory above all — where dimensionless quantities are the protagonists and conformal transformations are nonetheless decisive (Extra 6 of the previous series); there it is invariance under the transformation (a symmetry), not the transformation itself, that does the work. What is called "powerless" here is this notation, the operation of rewriting by a Weyl transformation.
Exercises (a wrap-up of Part II)
- What do all the quantities that "moved" in Part II have in common?
Show the answer
They are all dimensionful. \(G\), the Bohr radius, temperature, \(\rho_\Lambda\), viscosity, body mass — across nine fields, not one exception. Conversely, everything that did not move is dimensionless. - Name three quantities of weight \(+1\) and three of weight \(-1\).
Show the answer
\(+1\) (\(\div a\)): length, time, wavelength, Bohr radius, lifespan, Kolmogorov length. \(-1\) (\(\times a\)): mass, energy, temperature, frequency, gravitational acceleration, Lyapunov exponent. Sign and direction are opposite, which is where confusion creeps in. - In what kind of field is this notation powerless, and why?
Show the answer
Fields already written dimensionlessly — fluid similarity laws, critical phenomena, information theory, biological scaling. The tool touches only "size", so where nothing but dimensionless quantities appear it gives no information (Episode 13). Not a defect but the tool's definition. - Is "safe because dimensionless" an accurate statement?
Show the answer
No. As Episode 14 showed, the values of the weights themselves take quantum corrections (anomalous dimensions), measured to seven digits as \(\gamma_\sigma=0.0181489\) in the 3D Ising model. Precisely: "safe because observable" — measured dimensionless quantities do not move, but classically predicted weights can be wrong. - (Harder) Did Part II produce new physics? Then what was it for?
Show the answer
None at all. It rewrote known laws in a different notation and counted what moves. Its value, if any, is that the inventory "what does not move = physics" held without exception across nine fields from gravity to biology — and that the tool's range of application was measured precisely. It gives the previous series' decision procedure an actual service record.
Summary — only one thing ever moved
Part II put the same notation into nine fields: gravity, quantum mechanics, atoms, heat and information, light, the vacuum, fluids, phase transitions, chemistry and biology. The result always had the same shape — everything that moved was dimensionful, everything that did not was dimensionless. Across nine fields, not one exception.
The map of weights is complete too: number density at \(+3\); length, time and lifespan at \(+1\); velocity, \(\hbar\), entropy and every ratio at \(0\); mass, energy and temperature at \(-1\); energy density at \(-4\). And only the \(0\) column is observable.
What became clearest was the tool's range. A conformal transformation touches only "size", so it is powerful where size is the protagonist (cosmology, gravity) and completely powerless where shape is (fluids, critical phenomena, information, biology). Not a defect but the definition. With the same force that collapsed cosmology into "one mass" in Episode 4, Episode 13's Navier–Stokes equations did not collapse by a single character.
One proviso — as Episode 14 found, the weight table itself is a classical approximation carrying an error bar of \(\gamma_\sigma=0.0181489(10)\). So precisely, not "safe because dimensionless" but "safe because observable". Part V takes up the places where that distinction bites.
This document is Episode 16 of "c·t = const, That Clicks" (Part II wrap-up), written for physics-minded high-school and university readers. It summarises Episodes 7–15 and contains no new calculations — see the endnotes of each episode for its numbers and sources. The weight convention is \(\tilde X=\Omega^{w}X\) (\(\Omega=1/a\)), with lengths and times at \(w=+1\), mass, energy and temperature at \(-1\), and \(c,\hbar,e,\alpha\) and all dimensionless quantities at \(0\) — other sources use the opposite sign or write \(\Delta=-w\), so the convention must be checked before comparing. The "moved / did not move" classification applies to rewriting the whole universe at once and not to fixed laboratory conditions. The weight table is a classical approximation from dimensional analysis; in field theory an anomalous dimension is added, \(\Delta=\Delta_{\text{classical}}+\gamma\) (Episode 14; \(\gamma_\sigma=0.0181489(10)\) for the 3D Ising model) — so the accurate statement is "safe because observable", not "safe because dimensionless". The "helps / powerless" line is this series' own assessment; there are fields such as conformal field theory where dimensionless quantities are the protagonists and conformal invariance is nonetheless decisive (Extra 6 of the previous series) — what is called powerless here is the operation of rewriting by a Weyl transformation. Part II derives no new physics. 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).