c·t = CONST, THAT CLICKS EPISODE 8 ／ The equation does not change a letter. Only the mass does
Matter stops being able to walk; light keeps walking
Substituting into
quantum mechanics
The Schrödinger equation keeps its form — with \(m(t)=m_0\,t/t_0\).
And out of that: a free wave packet spreads only logarithmically.
Episode 7 substituted into gravity. This time, quantum mechanics. The conclusion first: the Schrödinger equation is not rewritten by a single character — except that the mass becomes \(m(t)=m_0\,t/t_0\). And out of that comes something rather hard to believe. A free wave packet spreads not in proportion to time, but only logarithmically. Integrate it up and you get one line: matter stops being able to walk; light keeps walking.
01First, check that the equation keeps its form
The weight of \(\psi\) is fixed by normalisation: \(\int|\psi|^2d^3x=1\) with \(d^3x\) of weight \(+3\) makes \(|\psi|^2\) weight \(-3\), so \(\psi\) is \(-3/2\). The rest is counting.
Left side \(i\hbar\,\partial\psi/\partial t\)
$$\underbrace{0}_{\hbar}+\underbrace{(-3/2)}_{\psi}+\underbrace{(-1)}_{\partial/\partial t}=-\frac52$$Right side \(\hbar^2\nabla^2\psi/2m\)
$$\underbrace{0}_{\hbar^2}-\underbrace{(-1)}_{1/m}+\underbrace{(-2)}_{\nabla^2}+\underbrace{(-3/2)}_{\psi}=+1-2-\frac32=-\frac52$$Conclusion of §01
Both sides come to \(-5/2\). The Schrödinger equation is unchanged to the letter in this picture too.
The only thing that changes is the mass inside it — \(m(t)=m_0\,t/t_0\).
This is the quantum-mechanical version of Episode 4's "delete everything deletable and one mass is left". Quantum mechanics has nothing but mass for a conformal transformation to catch.
02The heart — the wave packet spreads only logarithmically
Take a free particle. With no force the momentum is conserved and \(v=p/m\). But \(m\) grows in proportion to \(t\), so —
integrating
$$\Delta x(t)=\int v\,dt\ \propto\ \ln t$$| Standard picture | This picture | |
|---|---|---|
| Mass | constant | \(\propto t\) |
| Free particle velocity | constant | \(\propto 1/t\) |
| Wave packet spreading | \(\Delta x\propto t\) | \(\Delta x\propto\ln t\) |
You cannot notice this in a laboratory. \(\dot m/m=H_0\) means one e-fold in 13.8 billion years, utterly negligible for an electron wave packet that spreads in \(10^{-14}\) s. It matters only when integrated on cosmic scales.
03Integrating up — the reach of matter is finite
A particle set moving at time \(t_1\) with velocity \(v_1\), travelling forever after: how far does it get in comoving coordinates?
Anything with mass (velocity decaying as \(1/a\))
$$\Delta\chi=\int_{t_1}^{\infty}\frac{v_1(t_1/t)}{t/t_1}\,dt=v_1t_1^2\int_{t_1}^{\infty}\frac{dt}{t^2}=\boxed{\,v_1t_1\,}$$Light
$$\Delta\chi=\int_{t_1}^{\infty}\frac{c\,dt}{t/t_1}=c\,t_1\ln\frac{t}{t_1}\ \longrightarrow\ \infty$$The thing this episode most wants to say
Matter stops being able to walk. Light keeps walking.
The comoving reach of anything massive saturates at \(v_1t_1\) — the distance covered in the first Hubble time is the ceiling for all eternity.
| What sets out | Initial speed | Departure | Reach for all eternity, \(v_1t_1\) |
|---|---|---|---|
| Hydrogen atom at recombination | 5 km/s | 380,000 yr | 1.9 parsecs |
| Galaxy (today's peculiar velocity) | 600 km/s | today | 8.5 Mpc |
| Hot electron (today) | 1000 km/s | today | 14.1 Mpc |
| Light (today) | \(c\) | today | infinite (\(4230\,\mathrm{Mpc}\times\ln(t/t_0)\)) |
The first row is the one that bites. A hydrogen atom present at recombination can move only 2 comoving parsecs in the entire history of the universe. The pattern burned into the CMB stays frozen where it is, with nothing left but gravitational growth — and this integral is why.
Figure: how far a particle departing today ever reaches, in comoving distance. The matter curve always saturates; the light curve keeps climbing. The slider changes the initial speed, which moves only the height of the ceiling.
Push the slider all the way to \(c\) and the blue line still goes flat. Whether it saturates is not a question of speed — it is a question of having mass. With mass the velocity decays as \(1/a\) and the integral converges; without mass it does not decay and the integral diverges logarithmically.
04A table of what moves and what does not
| Quantity | Weight | In this picture |
|---|---|---|
| Uncertainty \(\Delta x\Delta p\ge\hbar/2\) | \(0\) | invariant |
| Action \(S/\hbar\) | \(0\) | invariant |
| Tunnelling probability \(e^{-2\int\kappa\,dx}\) | \(0\) | invariant |
| de Broglie wavelength \(h/p\) | \(+1\) | \(\div a\) |
| Compton wavelength \(\hbar/mc\) | \(+1\) | \(\div a\) |
| Thermal de Broglie wavelength \(h/\sqrt{2\pi mk_BT}\) | \(+1\) | \(\div a\) |
| Energy levels | \(-1\) | \(\times a\) |
Savour the third row. The tunnelling probability is completely invariant. Inside the exponent, \(\int\kappa\,dx\) has \(\kappa=\sqrt{2m(V-E)}/\hbar\) of weight \(-1\) and \(dx\) of weight \(+1\), so the product is \(0\). Therefore —
Conclusion of §04
The rate at which the Sun burns, the half-life of \(\alpha\) decay, the image in a scanning tunnelling microscope —
none of them change at all in this picture. All are fixed by dimensionless exponents alone.
05The reveal — the world grows more classical with time
So much is invariant, yet one dimensionless quantity moves steadily. It is the one from Episode 6 of the previous series.
today
$$N(\text{electron})=3.38\times10^{38},\qquad N(\text{hydrogen atom})=6.21\times10^{41}$$The standard picture reads this as "\(N\) grows because the universe grows". Here the universe is not expanding, so the reading changes — \(N\) grows because the Compton wavelength is shrinking. The ruler fattens and the quantum graininess gets finer.
Conclusion of §05
The universe becomes more classical with time.
\(N\propto t\) is a scale for how well a classical description works, and it alone cannot be moved by a conformal transformation.
── It is the ratio Episode 6 of the previous series meant by "the geometry vanished, the ratio survived".
① The Schrödinger equation is a non-relativistic approximation. The weights match on both sides provided \(V\) is transformed along with everything else as an energy (weight \(-1\)). An externally fixed potential (laboratory electrodes, say) does not transform by itself, and the story changes — what is treated here is rewriting the whole universe at once.
② "The packet spreads only as \(\ln t\)" is a statement about cosmological timescales. In the laboratory \(\dot m/m=H_0\simeq10^{-18}\)/s, beyond any measurement. And this is the same fact the standard picture states as "peculiar velocities decay as \(1/a\)" — not new physics.
③ The reach \(v_1t_1\) assumes \(a\propto t\) at all epochs. In the real universe (radiation → matter → \(\Lambda\)) the integral and its coefficients change — though the qualitative conclusion, massive things converge and light diverges, holds for any decelerating expansion. Read the table as an order-of-magnitude argument.
④ The 5 km/s for a recombination-era hydrogen atom is the thermal speed at \(T=3000\) K. Real baryons move collectively on acoustic waves, so treating them as free single particles is a coarse approximation.
⑤ The invariance of the tunnelling probability concerns the exponent in the WKB approximation. It is not a claim of complete invariance including prefactors and resonance conditions (though those too are invariant for the same reason wherever they can be written as dimensionless ratios).
Exercises (solvable with this episode's formulas alone)
- Why does \(\psi\) have weight \(-3/2\)?
Show the answer
From normalisation \(\int|\psi|^2d^3x=1\). Since \(d^3x\) has weight \(+3\), \(|\psi|^2\) has \(-3\), so \(\psi\) has \(-3/2\). The weight of the wave function is not chosen — normalisation fixes it. - How does a free particle's velocity change here, and what does that give for the wave packet?
Show the answer
No force, so \(p\) is conserved; mass \(\propto t\) gives \(v=p/m\propto1/t\). Integrating, \(\Delta x=\int v\,dt\propto\ln t\). Logarithmic, not linear. - Find the eternal comoving reach of a particle set moving at \(t_1\) with velocity \(v_1\).
Show the answer
\(\Delta\chi=\int_{t_1}^\infty v_1(t_1/t)/(t/t_1)\,dt=v_1t_1^2\int_{t_1}^\infty dt/t^2=v_1t_1\). The distance covered in the first Hubble time is the eternal ceiling. Only light diverges, as \(\int c\,dt/a\propto\ln t\). - Why is the tunnelling probability invariant?
Show the answer
Because the exponent \(\int\kappa\,dx\) is dimensionless: \(\kappa=\sqrt{2m(V-E)}/\hbar\) has weight \(-1\) (inverse length) and \(dx\) has \(+1\), summing to \(0\). The Sun's burning rate and \(\alpha\)-decay half-lives are untouched in this picture. - (Harder) With so much invariant, why does \(N=mc^2t/\hbar\) move?
Show the answer
\(N\) is "size of system ÷ Compton wavelength", and numerator and denominator draw their length from different places — \(t\) is the length the universe brings, \(\hbar/mc\) the length the particle brings. Strictly, in \(N=mc^2t/\hbar\) the \(m\) (\(-1\)) and \(t\) (\(+1\)) cancel, so \(N\) is invariant; but the fact that its value grows with time is the same in any gauge — as Episode 6 of the previous series confirmed by deriving the same formula from both pictures. Being invariant and being constant in time are different things.
Summary — matter stops walking, light keeps walking
Substituted into the Schrödinger equation, both sides come to weight \(-5/2\) and the equation is unchanged to the letter. Only the mass changes — \(m(t)=m_0t/t_0\). It is the quantum-mechanical version of Episode 4's "delete everything and one mass is left".
But that one thing bites. A free particle conserves momentum, so \(v=p/m\propto1/t\), and the packet spreads as \(\Delta x\propto\ln t\), not \(\propto t\). Integrated up, the comoving reach of anything massive saturates at \(v_1t_1\) — the distance covered in the first Hubble time is the eternal ceiling. For a hydrogen atom at recombination, 1.9 parsecs. That integral is why the CMB pattern stays frozen where it is. Light meanwhile diverges as \(c\,t_1\ln(t/t_1)\). Set out at light speed and you are overtaken once the universe is \(e=2.72\) times older.
We also counted the invariants: uncertainty, the action \(S/\hbar\), and tunnelling — since the exponent \(\int\kappa\,dx\) is dimensionless, the Sun's burning rate and \(\alpha\)-decay half-lives do not shift at all. What moves is de Broglie and Compton wavelengths and energy levels, differing only in whether they are a length or an energy.
And one quantity keeps rising: \(N=mc^2t/\hbar\) (\(3.4\times10^{38}\) for an electron). The standard picture says it rises because the universe grows; here it rises because the Compton wavelength shrinks. The universe is not expanding so much as growing coarse relative to the quantum graininess — the world becomes more classical with time.
This document is Episode 8 of "c·t = const, That Clicks", written for physics-minded high-school and university readers. That under a conformal transformation lengths and times carry weight \(+1\), mass and energy \(-1\), \(\hbar\) and \(c\) weight \(0\), and that normalisation gives the wave function \(-3/2\), is standard. That both sides of the Schrödinger equation come to weight \(-5/2\) so the equation keeps its form, and that a free particle's velocity decays as \(v\propto1/a\) (the standard redshifting of peculiar velocity), are also standard results. The \(\Delta\chi=v_1t_1\) — the finiteness of the comoving reach of a massive particle when \(a\propto t\) — together with the divergence of \(c\,t_1\ln(t/t_1)\) for light and the fact that even a particle setting out at \(c\) is overtaken at \(t=e\,t_1\), are calculated here. The table values (1.9 pc for a recombination-era hydrogen atom, 8.5 Mpc for a galaxy, 14.1 Mpc for an electron, \(c\,t_0=4230\) Mpc) are also computed here and assume \(a\propto t\) at all epochs — the coefficients change in the real universe, but the conclusion that massive things converge while light diverges holds for any decelerating expansion. The 5 km/s thermal speed at recombination is the \(T=3000\) K estimate; real baryons move collectively on acoustic oscillations. The invariance of the tunnelling probability is a claim about the exponent in the WKB approximation. That \(N=mc^2t/\hbar\) takes the same form in both pictures was shown in Episode 6 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).