Thinking in Waves Episode 7 ／ Part III · Physics as waves

The losing hypotheses stay on the page too

Scoring two conjectures ──
the neat 1/2 turned out to be an identity "Mass is the integral of the universe's gravity" ── compute it and $GM/(Rc^2)=0.500000$. Exactly one half.
But substitute the definitions and $G$, $c$ and $H$ all cancel. It was an identity.
And the testable version (Brans–Dicke) is ruled out at the $10^{-5}$ level.

Prerequisites: Episodes 4, 6Computation: kenshou/shitsuryo.js

This series began with two concrete conjectures ── "mass may be the integral of gravity over the age of the universe" and "the speed of light may be the integral of information".
Both are respectable directions. Let us compute and score them.

01Mach's principle ── a 150-year-old idea

1883

Machinertia is determined by the relation to all the matter in the universe

1953

Sciamaas a formula: $\sum_i Gm_i/(r_ic^2)\approx1$

"The sum of gravitational potentials from all the mass in the universe is of order $c^2$." Let us compute it.

Put the numbers in $$H_0=2.184\times10^{-18}\ \text{s}^{-1},\quad R=c/H_0=1.373\times10^{26}\ \text{m}\ (14.5\ \text{Gly})$$ $$\rho_c=\frac{3H_0^2}{8\pi G}=8.533\times10^{-27}\ \text{kg/m}^3,\quad M=\rho_c\cdot\frac43\pi R^3=9.241\times10^{52}\ \text{kg}$$ $$\boxed{\ \frac{GM}{Rc^2}=0.500000\ }$$

Exactly one half. Coincidence?

02The answer ── not a coincidence but an identity

Substitute and simplify $$M=\rho_c\cdot\frac43\pi R^3=\frac{3H^2}{8\pi G}\cdot\frac43\pi\frac{c^3}{H^3}=\frac{c^3}{2GH}$$ $$\frac{GM}{Rc^2}=\frac{G\cdot\dfrac{c^3}{2GH}}{\dfrac{c}{H}\cdot c^2}=\frac{c^3}{2H}\cdot\frac{H}{c^3}=\frac12\qquad\textbf{★ exactly}$$

The heart of §02

$G$, $c$ and $H$ all cancel.
The moment you define "a universe at critical density", this ratio is fixed at 1/2.

⇒ Sciama's coincidence is not evidence for Mach's principle. It is a restatement of flatness ── an identity.

It is not empty, though. That the universe is at critical density (flat) is an observational fact, and that is what makes the coincidence hold.

A textbook case of the "coincidence band"

Part III of the sister series "Understanding c·t = constant" classifies surprises in bits ── an identity is 0 bits, a coincidence 4–7 bits, something real is more.
This 1/2 scored exactly 0 bits. When a number comes out neat, first substitute everything and see whether it cancels ── that is the procedure.

03So can Mach's principle be tested? ── yes

The theory that turns "inertia is determined by the matter distribution" into something concrete is Brans–Dicke: replace $G$ by a field $\phi$ determined by the matter distribution.

Brans–Dicke $$\frac1G\ \longrightarrow\ \phi,\qquad \Box\phi=\frac{8\pi}{3+2\omega}T$$

$\omega\to\infty$ recovers general relativity. Finite $\omega$ means "mass is the integral of gravity" ── Machian

Measurement	Result	Meaning
Cassini spacecraft (2003)	$\omega>4\times10^{4}$	from the radio time delay
Deviation from GR	$\propto1/(2+\omega)<2.5\times10^{-5}$	bound on Machian effects

Verdict of §03

"Mass is the integral of gravity", once formalised and measured, is excluded at the $10^{-5}$ level.

⇒ The instinct pointed in a reasonable direction, but nature was not built that way.
The gravitational constant does not appear to be a quantity "computed from" the matter distribution.

04"Light speed = the integral of information" ── what can be computed

This too is a respectable direction. The holographic principle lets one compute how much information the universe can hold.

Holographic bound $$l_P=1.616\times10^{-35}\ \text{m},\qquad A=4\pi R^2=2.369\times10^{53}\ \text{m}^2$$ $$\text{maximum information}=\frac{A}{4l_P^2\ln2}=3.27\times10^{122}\ \text{bits}$$

Bound on the number of operations (Margolus–Levitin) $$\text{operation rate}\le\frac{2E}{\pi\hbar},\qquad E=Mc^2=8.31\times10^{69}\ \text{J}$$ $$\text{total operations}\le2.2\times10^{121}$$

The universe has performed $10^{121}$ operations on $10^{122}$ bits ── Lloyd's estimate. That much really can be computed.

05But ── "$c$ is the integral of information" does not follow

As we saw in Episode 4, $c$ is a defined value. Restating it in information language makes this even clearer.

Planck units $$l_P=1.616\times10^{-35}\ \text{m},\qquad t_P=5.391\times10^{-44}\ \text{s}$$ $$\frac{l_P}{t_P}=2.998\times10^{8}\ \text{m/s}=c\qquad(\text{exactly})$$

Verdict of §05

This is not a discovery; it is how Planck units are defined.
Even if spacetime were a computational lattice, $c$ would be "how many cells per tick" ── a rereading of the lattice's tick marks, not a quantity being integrated.

⇒ "$c$ is the maximum speed at which information travels" is correct.
But that is the definition of $c$, and does not make $c$ the integral of anything.

06So is there any "constant" that does come out of information? ── yes

An informational quantity that genuinely is an integral does exist: entropy.

Bekenstein–Hawking $$S=\frac{A}{4l_P^2}k_B,\qquad T=\frac{\hbar c^3}{8\pi GMk_B}$$

Mass	Temperature [K]	Evaporation time [yr]
solar mass	$6.17\times10^{-8}$	$2.10\times10^{67}$
Earth mass	$2.05\times10^{-2}$	$5.68\times10^{50}$
1 kg	$1.23\times10^{23}$	$2.67\times10^{-24}$
Planck mass	$5.64\times10^{30}$	$2.75\times10^{-47}$

The heart of §06

Mass fixes the temperature, temperature fixes the evaporation time. Not "constants" but a web of quantities linked to each other by integrals.

⇒ ★ The place where this line of thought pays off is not the constants themselves but the relations between them.

07Scoring

Claim	Verdict	Basis
audio derivative = high frequency	◎ exactly right	+6 dB/oct, measured in Episode 1
physics is a chain of calculus	◎ yes	an operation that keeps tilting the spectrum
constant of integration = physical constant	○ right	$\omega=0$ is undeterminable in principle
constants are integrals	◎ literally, for couplings	$\alpha(\mu)$ is the integral of the beta function; measured
differentiating a constant	△ depends on the axis	zero in time, non-zero in $\ln E$
mass = integral of gravity	× refuted	Brans–Dicke excluded at $10^{-5}$
light speed = integral of information	× does not hold	$c$ is defined; dimensionful quantities cannot be integrated into

Conclusion ── the hits and misses split along dimension

The framework "see everything as waves and link them by calculus" is right. But ──

Hits: dimensionless quantities ($\alpha$, $\mu$). These really are integrals, and really run.
Misses: dimensionful quantities ($c$, $\hbar$, $G$). They collapse into questions about the choice of units.

⇒ If you want to look further, look at dimensionless quantities. The Standard Model has about twenty of them, and not one has yet been derived as "the integral of" anything.

Exercises

When a beautifully round number appears, what should you suspect first?
Show answer
That it is an identity.
This episode's $GM/(Rc^2)=1/2$ is exactly that: substitute the definitions and $G, c, H$ all vanish. A relation in which the inputs never appear in the output has measured nothing.
The procedure: substitute every definition you used and check whether any symbol survives ── if none does, it is 0 bits.
Brans–Dicke was refuted, so why do people still talk about Mach's principle?
Show answer
Because what was refuted is one particular formalisation.
The idea that "inertia is relational" is partly absorbed into general relativity itself (coordinate-independent description, frame dragging / the Lense–Thirring effect). But the strong form, "$G$ is determined by the matter distribution", is excluded at $10^{-5}$.
── Separating the idea from the formalisation matters. Only the latter can be scored.
Why is "light speed is the maximum speed of information" right while "light speed is the integral of information" is wrong?
Show answer
The first explains what $c$ means; the second claims where $c$ comes from.
Since 1983 $c$ is a defined value, so asking where it comes from lands you at the definition of the metre. Only dimensionless quantities admit origin questions ── "why $\alpha\approx1/137$?" is a genuine open question nobody has answered.
Episode 4's "dimensionful is bookkeeping, dimensionless is physics" is at work here too.

Summary　Two losses and one win

Mass = the integral of gravity: Sciama's relation gives $GM/(Rc^2)=0.500000$ ── exactly 1/2. But substituting $M=c^3/(2GH)$ makes $G$, $c$ and $H$ all vanish; in a critical-density universe it is an identity. And the testable form (Brans–Dicke) is bounded by Cassini at $\omega>4\times10^4$, excluding it at the $10^{-5}$ level.

Light speed = the integral of information: the holographic bound is computable ($3.3\times10^{122}$ bits, $2.2\times10^{121}$ operations). But $l_P/t_P=c$ is a definition, not a discovery ── $c$ is a rereading of the lattice spacing, not something being integrated.

Where the idea pays off is not the constants but the relations between them ── as with a black hole's temperature and evaporation time.

The hits and misses split along dimension.
Dimensionless quantities really are integrals that run; dimensionful ones collapse into unit conventions.

Next time ── the summing-up We fold seven episodes onto one page. What physics can be written in wave language, and what cannot.
And we count the doors still open ── the questions this series found that nobody has closed.

This document is Episode 7 of the series "Thinking in Waves". The computation is in kenshou/shitsuryo.js. On the numbers: we used $H_0=67.4$ km/s/Mpc (the Planck value) ── because of the Hubble tension, a local value (around 73) changes $R$ and $M$, but $GM/(Rc^2)=1/2$ is an identity and is unaffected (which is the whole point of that section). The Brans–Dicke bound $\omega>4\times10^4$ is the published Cassini Shapiro-delay result, not computed here. The holographic bound and Lloyd's operation count both rest on the strong assumption of treating the interior of a horizon as one system ── the numbers change with the choice of horizon (Hubble / particle / event). The black-hole evaporation times use the standard Schwarzschild formulas; in reality absorption from the CMB means that solar-mass holes do not evaporate in the present universe (they are colder than the CMB). ── To print, use your browser's Print → Save as PDF.

Mass	Temperature [K]	Evaporation time [yr]
solar mass	\(6.17\times10^{-8}\)	\(2.10\times10^{67}\)
Earth mass	\(2.05\times10^{-2}\)	\(5.68\times10^{50}\)
1 kg	\(1.23\times10^{23}\)	\(2.67\times10^{-24}\)
Planck mass	\(5.64\times10^{30}\)	\(2.75\times10^{-47}\)