c·t = CONST, THAT CLICKS EPISODE 13 ／ Leaving cosmology entirely
Every dimensionless number is invariant — so what does the breakdown look like?
Substituting into
fluids and turbulence
The four quantities in the Reynolds number move as \(a^4,\ a^0,\ a^{-1},\ a^3\), all differently.
And combined, they always come back to zero.
From here the second half of Part II leaves cosmology entirely. Today: fluids and turbulence. Every dimensionless number engineering uses — Reynolds, Mach, Prandtl, Froude — is invariant without exception. So similarity laws, wind tunnel testing and Kolmogorov's \(-5/3\) law are unchanged to the letter here. The interesting part is the breakdown: the four quantities that make up the Reynolds number move with completely different weights, and always come back to zero when combined.
01First, count the weights mechanically
The procedure is simple. Any quantity decomposes as "length\(^{n_L}\) × time\(^{n_T}\) × mass\(^{n_M}\)", so substitute length \(+1\), time \(+1\), mass \(-1\) and add.
| Quantity | Dimensions | Weight | In this picture |
|---|---|---|---|
| Length \(L\) | \(L\) | \(+1\) | \(\div a\) |
| Velocity \(v\) | \(L/T\) | \(0\) | invariant |
| Mass density \(\rho\) | \(M/L^3\) | \(-4\) | \(\times a^4\) |
| Pressure \(p\) | \(M/LT^2\) | \(-4\) | \(\times a^4\) |
| Viscosity \(\eta\) | \(M/LT\) | \(-3\) | \(\times a^3\) |
| Kinematic viscosity \(\nu=\eta/\rho\) | \(L^2/T\) | \(+1\) | \(\div a\) |
| Dissipation rate \(\varepsilon\) | \(L^2/T^3\) | \(-1\) | \(\times a\) |
| Surface tension \(\sigma\) | \(M/T^2\) | \(-3\) | \(\times a^3\) |
Look at the second row — velocity has weight 0. Length and time share a weight, so they cancel in the quotient. This is the plainest reason behind everything this series has said about \(c\) not moving.
02Run every dimensionless number through it
| Dimensionless number | Definition | Sum of weights | Result |
|---|---|---|---|
| Reynolds number | \(\rho vL/\eta\) | \(-4+0+1-(-3)=0\) | invariant |
| Mach number | \(v/c_s\) | \(0-0=0\) | invariant |
| Prandtl number | \(\nu/\kappa\) | \(1-1=0\) | invariant |
| Froude number | \(v/\sqrt{gL}\) | \(0-\tfrac{-1+1}{2}=0\) | invariant |
| Weber number | \(\rho v^2L/\sigma\) | \(-4+0+1-(-3)=0\) | invariant |
| Strouhal number | \(\Omega L/v\) | \(-1+1-0=0\) | invariant |
| Kolmogorov length | \((\nu^3/\varepsilon)^{1/4}\) | \((3\cdot1-(-1))/4=1\) | same as a length (\(\div a\)) |
Conclusion of §02
Every dimensionless number of fluid mechanics is invariant.
So similarity laws hold unchanged — wind tunnels and ship model basins work exactly as before in this picture.
03The heart — the breakdown is thrashing about
how each one moves
$$\tilde\rho=a^4\rho,\qquad \tilde v=v,\qquad \tilde L=\frac{L}{a},\qquad \tilde\eta=a^3\eta$$assembled
$$\widetilde{\mathrm{Re}}=\frac{(a^4\rho)(v)(L/a)}{a^3\eta}=a^{4-1-3}\,\frac{\rho vL}{\eta}=\mathrm{Re}$$The four parts move by \(a^4\), \(a^0\), \(a^{-1}\), \(a^3\) — utterly different factors — and the exponents come to exactly \(4-1-3=0\). Not a coincidence, of course: it restates that the Reynolds number is dimensionless. But watching the restatement happen is quite a sight.
Figure: how much each part of the Reynolds number moves in this picture. Drag back in time and the bars stretch and shrink independently (\(\rho\) as \(a^4\), \(\eta\) as \(a^3\), \(L\) as \(1/a\)). And the rightmost bar — the total, Re — stays pinned to zero.
04Turbulence and critical phenomena too — exponents are just numbers
| What is measured | Value | In this picture |
|---|---|---|
| Kolmogorov law \(E(k)\propto k^{-5/3}\) | \(-5/3\) | invariant |
| 3D Ising \(\nu\) | 0.629971 | invariant |
| Scaling dimension \(\Delta_\sigma\) | 0.5181489 | invariant |
| Fractal dimension | ── | invariant |
| Lyapunov exponent × elapsed time \(\lambda t\) | ── | invariant |
| Kleiber's law exponent | 3/4 | invariant |
The Lyapunov exponent \(\lambda\) alone has weight \(-1\) on its own (inverse time), but what has meaning is the product with elapsed time, \(\lambda t\). Since \(t\) has weight \(+1\), it cancels. "How much predictability has been lost" has the same value in this picture.
05The reveal — the tool only touches "size"
The thing this episode most wants to say
A conformal transformation touches only "size".
Every measure of structure, complexity and information is dimensionless — out of reach.
This is the complex-systems version of Episode 6's conclusion. There we counted "it cannot reach the memory in use (= black hole entropy)". Here: "it cannot reach turbulence exponents, critical exponents or fractal dimensions either".
So while cosmology collapsed into "one mass" under this notation (Episode 4), fluid mechanics does not collapse by a single character. There is nothing in it to collapse. The Navier–Stokes equations are completely untouched here.
① The weight counting assumes material properties (viscosity and the like) are transformed along with everything else. A fluid fixed in a laboratory (water of a given viscosity) does not transform by itself — what is treated here is rewriting the whole universe at once. Same caveat as Episode 8 ①.
② "Wind tunnels still work" restates the fact that similarity laws are written with dimensionless numbers only. Real wind tunnel testing works independently of conformal transformations; this document merely reconfirms that and claims nothing new.
③ Kolmogorov's \(-5/3\) law has intermittency corrections and deviates slightly from \(-5/3\) in reality. The corrected exponent is also dimensionless, so the conclusion is unchanged.
④ "A conformal transformation touches only size" is a classical statement. Quantised, the trace anomaly makes dimensionless quantities (couplings, scaling dimensions) run — treated in Episode 8 and Extra 6 of the previous series, and in Part V here. Dimensionless does not mean absolutely safe.
Exercises (solvable with this episode's formulas alone)
- Why does velocity have weight 0?
Show the answer
Because length and time both have weight \(+1\), so they cancel in the quotient. This is the plainest reason the speed of light does not move in this picture — \(c\) is a velocity, hence weight 0 from the start. - Find the weight of viscosity \(\eta\) (dimensions \(M/LT\)).
Show the answer
\(M\) is \(-1\), \(L\) is \(+1\), \(T\) is \(+1\), so \(-1-1-1=-3\), giving \(\tilde\eta=a^3\eta\). The younger the universe, the runnier the fluid in this picture. - Show the Reynolds number is invariant from the four parts' exponents.
Show the answer
\(\tilde\rho=a^4\rho\), \(\tilde v=v\), \(\tilde L=L/a\), \(\tilde\eta=a^3\eta\). Assembled: \(a^{4}\cdot a^{0}\cdot a^{-1}/a^{3}=a^{0}\). Four parts moving independently, exponents summing to zero. - What is the weight of the Kolmogorov length \((\nu^3/\varepsilon)^{1/4}\), and what does it mean?
Show the answer
\(\nu\) is \(+1\) and \(\varepsilon\) is \(-1\), so \((3\cdot1-(-1))/4=+1\) — the same weight as a length. The Kolmogorov scale shrinks as \(1/a\) like every other length here. The hierarchy of eddies shrinks similarly, whole, without changing shape. - (Harder) Does this series' tool say anything about complex systems?
Show the answer
Nothing at all. Every measure of complexity (dimensionless numbers, exponents, fractal dimensions, information measures) has weight 0, so a conformal transformation moves none of them. This tool touches only "size" and cannot reach "shape" — the same structure as Episode 6's "it cannot reach the memory in use". And that is a precise statement of the tool's limit.
Summary — it touches size and cannot reach shape
Decompose any quantity as "length\(^{n_L}\)×time\(^{n_T}\)×mass\(^{n_M}\)", add \(+1,+1,-1\), and out comes the weight. Doing so: velocity 0, mass density \(-4\), viscosity \(-3\), kinematic viscosity \(+1\). And Reynolds, Mach, Prandtl, Froude, Weber and Strouhal numbers are all 0 without exception. Similarity laws hold unchanged; wind tunnels and ship model basins work exactly as before.
The breakdown was the interesting part. The Reynolds number's four parts move by \(a^4\), \(a^0\), \(a^{-1}\), \(a^3\) — utterly different factors — and the exponents sum to \(4-1-3=0\) exactly. That the result is invariant is obvious; watching the innards thrash about this much is a sight.
A level up is the same: Kolmogorov's \(-5/3\), the 3D Ising \(\nu=0.629971\), the scaling dimension \(\Delta_\sigma=0.518\), fractal dimensions, Lyapunov exponent × elapsed time, Kleiber's \(3/4\). Exponents are just numbers, so not one of them moves.
And the reveal — a conformal transformation touches only "size". Measures of structure, complexity and information are all dimensionless and out of reach. So while cosmology collapsed to "one mass" (Episode 4), fluid mechanics does not collapse by a character. Navier–Stokes stands completely untouched. Which is also a precise measurement of the tool's limit: a conformal transformation says something new only where dimensionful quantities are the protagonists — almost exclusively cosmology and gravity.
This document is Episode 13 of "c·t = const, That Clicks", written for physics-minded high-school and university readers. That lengths and times carry conformal weight \(+1\) and mass \(-1\), so that any quantity's weight follows mechanically from its dimensions, is standard. The weights given here (velocity 0, mass density \(-4\), pressure \(-4\), viscosity \(-3\), kinematic viscosity \(+1\), dissipation rate \(-1\), surface tension \(-3\)), the vanishing weight of the Reynolds, Mach, Prandtl, Froude, Weber and Strouhal numbers, and the weight \(+1\) of the Kolmogorov length, are mechanical checks performed here (kenshou/calc18.py). They are instances of the general rule that dimensionless numbers are conformally invariant and claim no new physics — similarity laws and wind tunnel testing hold independently of conformal transformations, and this document merely reconfirms that. The weight counting assumes material properties (viscosity and so on) transform along with everything else and does not apply to a fluid fixed in a laboratory. Kolmogorov's \(-5/3\) law has intermittency corrections and deviates slightly in reality; the corrected exponent is also dimensionless, so the conclusion is unchanged. The 3D Ising values \(\nu=0.629971\) and \(\Delta_\sigma=0.5181489\) are conformal bootstrap results (Extra 6 of the previous series). "A conformal transformation touches only size" is a classical claim; quantised, the trace anomaly makes couplings and scaling dimensions run (Episode 8 and Extra 6 of the previous series; Part V here). 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).