c·t = CONST, THAT CLICKS EPISODE 15 ／ Part II, at the farthest remove
Everything life can measure turned out to be dimensionless
Substituting into
chemistry and biology
The Arrhenius factor, Kleiber's exponent, heartbeats per lifetime, the information in DNA — all invariant.
So life has no way of knowing which picture it is in.
Part II closes at the farthest remove — chemistry and biology. Bond energy ÷ \(k_BT\), the Arrhenius factor, pH, Kleiber's exponent, heartbeats per lifetime, the information content of DNA. All dimensionless without exception, so nothing changes by a character in this picture. And the conclusion that follows is a strong one: life cannot, in principle, know whether the universe is "expanding" or "growing in mass".
01Chemistry comes through entirely intact
\(E_a\) and \(k_BT\) both have weight \(-1\), so the ratio is invariant
| Chemical quantity | Content | In this picture |
|---|---|---|
| Arrhenius factor | \(e^{-E_a/k_BT}\) | invariant |
| Equilibrium constant | \(e^{-\Delta G/RT}\) | invariant |
| pH | log of a concentration ratio | invariant |
| Bond energy ÷ thermal energy | \(E_b/k_BT\) | invariant |
| Ratios of reaction rates | ── | invariant |
Exactly the reason tunnelling was invariant in Episode 8 — the exponent is dimensionless. So the Sun's burning rate and enzyme kinetics are both untouched here.
02Taking Kleiber's law apart
The most famous scaling law in biology — metabolic rate goes as body mass to the 3/4, holding over six orders of magnitude from mouse to elephant.
Metabolic rate \(B\) (energy per time = \(ML^2/T^3\))
$$w(B)=-1+2\cdot1-3\cdot1\cdot(-1)\ \Longrightarrow\ w=-2\qquad(\times a^2)$$Body mass \(M\)
$$w(M)=-1\qquad(\times a)$$For \(B=CM^{3/4}\) to keep its form
$$w(C)=-2-\tfrac34(-1)=-\tfrac54\qquad(\times a^{5/4})$$Conclusion of §02
The exponent 3/4 is invariant. The coefficient \(C\) moves as \(\times a^{5/4}\).
The coefficient is dimensionful — bookkeeping. The gauge-invariant statement is "the ratio of two animals' metabolic rates = (mass ratio)\(^{3/4}\)", a ratio of ratios, which does not move.
The surgery of Episode 3 (on the series title), Episode 9 (on the atom) and Episode 12 (on the cosmological constant) applies here in the same shape — the 3/4 in "metabolic rate goes as mass to the 3/4" is physics; the coefficient is bookkeeping.
03Heartbeats per lifetime were dimensionless
so the product is
$$\text{heartbeats per lifetime}\ \propto M^{0}\ \simeq\ 1.5\times10^{9}$$Mouse and elephant alike, roughly 1.5 billion beats in a lifetime. Independent of mass — that is, dimensionless. So the same 1.5 billion here.
04Listing everything life can measure
Figure: the top row is dimensionful quantities (mass, length, metabolic rate, lifespan) — drag back in time and they thrash. The bottom row is what life actually measures — every one pinned to 1.
| What life can measure | Content | In this picture |
|---|---|---|
| Concentration ratios | count ÷ count | invariant |
| Ratios of reaction rates | time ÷ time | invariant |
| Heartbeats per lifetime | a count | invariant (\(1.5\times10^9\)) |
| Number of generations | a count | invariant |
| Genetic information | bits | invariant (\(6.2\times10^9\) bit = 775 MB) |
| Body length ÷ cell size | length ÷ length | invariant |
| Ratios of metabolic rates | ── | invariant |
| Temperature difference ÷ temperature | ── | invariant |
The thing this episode most wants to say
Everything life can measure is dimensionless, without exception.
So life cannot, in principle, know whether the universe is "expanding" or "growing in mass".
Episode 9 said "an atom has no comparison partner". This pushes that up to the scale of biology. Cells, hearts and genes cannot notice the difference between the pictures, so long as their comparisons bottom out in atoms.
05An aside — two kinds of "step count"
| Steps of what | Count | From |
|---|---|---|
| The universe's logarithmic clock | 140.2 | Episode 2 |
| The universe's Planck ticks | \(8.08\times10^{60}\) | Episode 1 |
| A human's lifetime heartbeats | \(1.5\times10^{9}\) | today |
| Information in the human genome | \(6.2\times10^{9}\) bit | today |
| The universe's memory | \(2.96\times10^{122}\) bit | Episode 1 |
The third row is the fun one — the universe has made only 140 moves, while a human heart beats 1.5 billion times in a lifetime. The units of counting differ, of course (doublings versus beats). Still, both are dimensionless and neither moves when the picture is swapped.
① Kleiber's exponent 3/4 is disputed. Some analyses support 2/3 (a surface-area law), and the effective exponent varies with taxon and mass range. What is used here is only that whatever the exponent is, it is dimensionless and therefore invariant; the value 3/4 itself is not being asserted.
② "\(1.5\times10^9\) heartbeats per lifetime" is a rough mammalian marker. Species vary by factors of a few, and humans (partly thanks to medicine) sit above it. What matters here is the structure of being mass-independent, hence dimensionless, not the precision of the number.
③ It is assumed that chemical and biological quantities transform along with everything else. Same caveat as Episode 8 ① and Episode 13 ①: fixed laboratory conditions (a thermostat at a set temperature) change the story — what is treated here is rewriting the whole universe at once.
④ "Life cannot know in principle" is a consequence of this series' decision procedure. More precisely: so long as only dimensionless quantities are measured, the two cannot be distinguished — a limitation not peculiar to life but bearing equally on every observer (final episode of the previous series).
⑤ The human genome's \(3.1\times10^9\) base pairs × 2 bit is an upper bound for a naive encoding. The actual information content (compressed, or functionally meaningful) is smaller, and estimating it is contested.
Exercises (solvable with this episode's formulas alone)
- Why is the Arrhenius factor invariant here?
Show the answer
Because the exponent \(E_a/k_BT\) is energy ÷ energy = dimensionless. Exactly the reason tunnelling was invariant in Episode 8, and reaction rates are untouched. - Find the weight of metabolic rate \(B\) (dimensions \(ML^2/T^3\)).
Show the answer
\(M\) is \(-1\), \(L^2\) is \(+2\), \(T^{-3}\) is \(-3\), giving \(-1+2-3=-2\), so \(\tilde B=a^2B\). Older creatures have slower metabolism in this picture — but so does everything they are compared with. - How does the coefficient \(C\) move in Kleiber's law \(B=CM^{3/4}\)?
Show the answer
\(w(C)=w(B)-\tfrac34w(M)=-2-\tfrac34(-1)=-\tfrac54\), i.e. \(\times a^{5/4}\). The exponent 3/4 is invariant; the coefficient moves — being dimensionful, it is bookkeeping. The gauge-invariant statement is "the ratio of two animals' metabolic rates = (mass ratio)\(^{3/4}\)". - Show that heartbeats per lifetime are mass-independent, and say what happens here.
Show the answer
Heart rate \(\propto M^{-1/4}\) and lifespan \(\propto M^{+1/4}\), so the product is \(M^0\). Mass-independent, hence dimensionless, so the same \(1.5\times10^9\) in this picture. Lifespan shrinks as \(\div a\) and the heart speeds up by the same factor, leaving the count unchanged. - (Harder) Is "life cannot know which picture it is in" a limitation peculiar to life?
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No — it bears equally on every observer. "Only dimensionless quantities are measurable" is the decision procedure of the previous series' final episode. Biology was brought in to see that even the things furthest from cosmology — cells, hearts, genes — give the same conclusion. It is Episode 9 (the atom) pushed up to biological scale.
Summary — life has no way of knowing which picture it is in
Chemistry came through entirely intact — Arrhenius factor, equilibrium constant, pH, bond energy ÷ \(k_BT\). The exponents are dimensionless, so reaction rates do not change at all (the same reason as tunnelling in Episode 8).
In biology we took Kleiber's law apart. Metabolic rate has weight \(-2\) (\(\times a^2\)) and body mass \(-1\) (\(\times a\)), so in \(B=CM^{3/4}\) the coefficient moves as \(\times a^{5/4}\) while the exponent 3/4 is invariant. The gauge-invariant statement is "the ratio of two animals' metabolic rates = (mass ratio)\(^{3/4}\)" — exactly the surgery of Episodes 3, 9 and 12.
Then heartbeats per lifetime. Heart rate \(\propto M^{-1/4}\) and lifespan \(\propto M^{1/4}\), so the product is \(M^0\): about 1.5 billion, independent of mass. Dimensionless, hence the same value here. Measure a life in seconds and it shrinks; measure it in heartbeats and it does not — the same structure as Episode 9's Bohr radius.
List everything life can measure — concentration ratios, rate ratios, heartbeats, generations, bits of genetic information, body length ÷ cell size, metabolic ratios, temperature difference ÷ temperature. Dimensionless without exception. So life cannot, in principle, know whether the universe is "expanding" or "growing in mass". Episode 9's "an atom has no comparison partner", pushed up to biological scale. And as an aside — the universe has made only 140 moves, while a human heart beats 1.5 billion times in a lifetime.
This document is Episode 15 of "c·t = const, That Clicks", written for physics-minded high-school and university readers. That dimensionless quantities are conformally invariant, and that weights follow from decomposing dimensions, are standard (Episode 13). The metabolic-rate weight \(-2\) and the Kleiber coefficient weight \(-5/4\) are computed here (kenshou/calc20.py). Kleiber's exponent 3/4 is disputed — some analyses support 2/3 (a surface-area law), and the effective exponent varies with taxon and mass range — so what is used here is only that whatever the exponent is, it is dimensionless and therefore invariant. "\(1.5\times10^9\) heartbeats per lifetime" is a rough mammalian marker with species-level variation of a few, and humans sit above it; what matters is the mass-independent, hence dimensionless, structure, not the precision. The human genome's \(3.1\times10^9\) base pairs × 2 bit \(=6.2\times10^9\) bit is an upper bound for a naive encoding; the actual (compressed or functional) content is smaller and contested. Chemical and biological quantities are assumed to transform along with everything else, so this does not apply to fixed laboratory conditions (same caveat as Episode 8 ① and Episode 13 ①). "Life cannot know in principle" means, more precisely, "so long as only dimensionless quantities are measured, the two cannot be distinguished" — a limitation on every observer, not peculiar to life (final episode of the previous series). 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).