c·t = CONST, THAT CLICKS EPISODE 11 ／ The photon gas is completely at rest
Light had no room to stretch in the first place
Substituting
into light
Number density, energy density, temperature, wavelength — all constant.
Nothing has happened to light in the entire history of the universe.
We have written many times that light passes straight through a conformal transformation, but never treated it head on. Today we do. Look at a photon gas in this picture and — number density, energy density, temperature, wavelength, all constant. Completely at rest. Nothing has happened to light in the entire history of the universe; only matter has been growing. The result of Episode 7 of the previous series — that only the 4D Maxwell action is exactly conformally invariant — shows itself in its most naked form.
01Transforming the photon gas, item by item
| Quantity | Weight | Standard picture | This picture |
|---|---|---|---|
| Photon number density \(n_\gamma\) | \(+3\) | \(\propto a^{-3}\) | constant |
| Energy density \(\rho_\gamma\) | \(-4\) | \(\propto a^{-4}\) | constant |
| Temperature \(T\) | \(-1\) | \(\propto a^{-1}\) | constant |
| Energy of one photon \(\hbar\omega\) | \(-1\) | \(\propto a^{-1}\) | constant |
| Wavelength \(\lambda\) | \(+1\) | \(\propto a\) | constant |
Conclusion of §01
Nothing about light moves at all.
Today's photon number density \(4.11\times10^{8}\ \mathrm{m^{-3}}\) is the same value throughout the history of the universe.
The photon gas in this picture is completely at rest.
This is no accident. The power of \(a\) by which each photon quantity varies in the standard picture, and that quantity's weight, are exactly the same number. Multiply and they must cancel. Light is built to fit a conformal transformation with nothing left over.
02And not one observation changes
"Everything about light is constant" sounds as if something must break. Nothing does — because only two dimensionless numbers fix the CMB.
① entropy per photon
$$\frac{s_\gamma}{n_\gamma}=\frac{1.478\times10^{9}}{4.111\times10^{8}}=3.60\ k_B\qquad(\text{blackbody theory: }3.602)$$② baryon-to-photon ratio
$$\eta=\frac{n_b}{n_\gamma}=6.1\times10^{-10}$$Both are dimensionless, hence identical in this picture and the standard one. And the blackbody spectral shape is fixed by the dimensionless combination \(\hbar\omega/k_BT\) alone, so it too is invariant. The Planck distribution is unchanged to the letter.
03Redshift turns completely inside out
If nothing happens to light, where does redshift come from? From the receiver.
| Standard picture | This picture | |
|---|---|---|
| Light in flight | space stretches, so the wavelength stretches | nothing happens |
| Laboratory hydrogen atom | always the same | getting heavier (\(\tilde m=am\)) |
| Reference Lyman α | always the same wavelength | gets shorter with time |
| What the spectrograph reads, \(1+z\) | \(\lambda_{\rm obs}/\lambda_{\rm lab}\) | the same value |
Lyman α from a galaxy at \(z=7\)
$$\text{standard: the wavelength stretched }8\times\qquad\text{here: the laboratory reference got }8\times\text{ finer}$$The CMB at \(z=1100\)
$$1101\times\qquad\text{either reading gives the same spectrograph scale}$$Not "the light stretched" but "the ruler grew" — what Episodes 2 and 3 of the previous series said in words is now fully backed by §01's table. There is no room on the light's side for anything to stretch.
Figure: the slider switches the way of speaking. At the left (standard) the photon quantities scatter in all directions; at the right (this picture) all four go perfectly flat. The dimensionless ratios (\(s/n\) and \(\eta\)) never move anywhere.
04The reveal — it balances exactly, only in four dimensions
the volume element produces \(D\) factors of \(\Omega\); two inverse metrics eat 4
$$S_{\rm EM}\ \longrightarrow\ \Omega^{\,D-4}\,S_{\rm EM}$$| Dimension \(D\) | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|
| Residual factor | \(\Omega^{-2}\) | \(\Omega^{-1}\) | \(\Omega^{0}=1\) | \(\Omega^{+1}\) | \(\Omega^{+2}\) |
What is produced balances what is eaten only at \(D=4\). That we live in four dimensions is why the photon gas looks stationary here. In five dimensions, light would move even in this picture, and "push the whole expansion into mass" would fail.
05The plainer statement — light has nothing to compare against
Episode 3 said "you need a comparison to call \(c\cdot t\) constant"; Episode 9 said "you need a comparison to say atoms are shrinking". Light has no comparison partner at all — so it has nothing to say about a conformal transformation. That is the everyday translation of "conformally invariant".
06Except that quantum theory breaks it
All of the above is classical. Quantise and conformal symmetry breaks even in a massless theory, because defining a field theory requires a scale \(\mu\) (Episode 8 of the previous series).
measured in experiment
$$\frac{1}{\alpha}:\ 137.036\ \longrightarrow\ 127.95\quad(\text{from the electron mass to }M_Z)$$So "nothing happens to light" is a classical statement. Quantum theory forces a ruler onto light after all. Part V (Episode 37) treats this breaking head on — the first of the places where this series' tool breaks.
① "The photon gas is at rest" refers to the coordinates after the conformal transformation. The locally measured speed of light is \(c_0\) in every picture, and a CMB thermometer reads 2.7255 K in every picture. No observation changes when the picture is swapped.
② \(n_\gamma\propto a^{-3}\) and \(T\propto a^{-1}\) hold for adiabatic expansion. During epochs when annihilation dumps entropy there are corrections from the change in \(g_{*s}\) (Episode 2 computed 1.10 nat). The table shows the plain dependence with those corrections removed.
③ "Nucleosynthesis is a function of \(\eta\) alone" is a simplification. Strictly, the effective relativistic degrees of freedom \(g_*\), the neutron lifetime, and the expansion rate also matter — and it was the expansion rate that ruled out \(a\propto t\) in Extra 1 of the previous series. The claim here is only that swapping pictures does not move \(\eta\), so the verdict does not change because of the picture.
④ \(s_\gamma/n_\gamma=3.60\) is computed here from the numbers; the exact blackbody value is \(4\pi^4/(45\zeta(3))=3.6017\). The \(n_\gamma=4.11\times10^8\ \mathrm{m^{-3}}\) and \(s_\gamma=1.478\times10^9\ k_B\,\mathrm{m^{-3}}\) used are standard values at \(T_0=2.7255\) K.
⑤ The \(\Omega^{D-4}\) counting takes \(A_\mu\) (lower index) to have weight 0. That is the standard convention; writing \(A^\mu\) obviously changes the weight — whether the action is invariant does not depend on how you write it, but the intermediate bookkeeping does.
Exercises (solvable with this episode's formulas alone)
- Explain by weights why the photon number density is constant here.
Show the answer
\(n_\gamma\) has weight \(+3\) (inverse volume), so \(\tilde n=a^3n\); in the standard picture \(n\propto a^{-3}\). Multiplying, \(a^3\cdot a^{-3}=1\) — exact cancellation. Energy density (\(-4\) with \(a^{-4}\)), temperature (\(-1\) with \(a^{-1}\)) and wavelength (\(+1\) with \(a\)) have the same structure. - Name the two dimensionless numbers that fix the CMB and say what happens to them here.
Show the answer
Entropy per photon \(s/n=3.60\,k_B\) and the baryon-to-photon ratio \(\eta=6.1\times10^{-10}\). Both are dimensionless, so they do not move at all. The blackbody shape depends only on \(\hbar\omega/k_BT\) and is likewise invariant. - In what dimension is the Maxwell action conformally invariant, and why?
Show the answer
\(D=4\) only. \(\sqrt{-g}\) produces \(D\) factors of \(\Omega\) and two inverse metrics eat 4, leaving \(\Omega^{D-4}\). That we live in four dimensions is why the photon gas looks stationary in this picture. - Explain the Lyman α of a \(z=7\) galaxy in both pictures.
Show the answer
Standard: space stretched 8-fold, so the wavelength in flight stretched 8-fold. Here: the light arrives unchanged, and laboratory hydrogen got heavier so the reference Lyman α became 8 times finer. The spectrograph reads \(\lambda_{\rm obs}/\lambda_{\rm lab}=8\) either way. - (Harder) How far is "nothing happens to light" correct?
Show the answer
As far as classical physics. Quantisation requires a scale \(\mu\), so conformal symmetry breaks even in a massless theory (the trace anomaly \(T^\mu{}_\mu=(\beta/2g)F^2\)). The breaking is measured: \(1/\alpha\) running from 137.036 to 127.95. Quantum theory forces a ruler onto light after all.
Summary — light had no room to stretch
Transform the CMB quantities by the weight table and number density, energy density, temperature, per-photon energy and wavelength are all constant, because the power of \(a\) in the standard picture and the weight of each quantity are exactly the same number. The photon gas in this picture is completely at rest — today's \(4.11\times10^8\ \mathrm{m^{-3}}\) is its value throughout cosmic history.
And no observation changes. Only two dimensionless numbers fix the CMB — entropy per photon \(3.60\,k_B\) and the baryon-to-photon ratio \(\eta=6.1\times10^{-10}\). Neither moves, and the blackbody shape depends only on \(\hbar\omega/k_BT\). Since the nucleosynthesis prediction is a function of \(\eta\), swapping pictures cannot move the verdict — which is what Extra 1's "defend it in any of the three pictures, same result" came down to.
And redshift turns inside out. With no room on the light's side, what changed is the receiver. At \(z=7\), not "the wavelength stretched 8-fold" but "the laboratory reference got 8 times finer". The spectrograph reads the same 8.
The reveal was dimensional: the Maxwell action picks up \(\Omega^{D-4}\), so it is exactly conformally invariant only at \(D=4\). That we live in four dimensions is why this picture works at all. More plainly, light carries neither ruler nor clock (infinite Compton wavelength, zero proper time) — so when the ruler is swapped it has nothing to compare against and cannot notice. All classical, though: quantise and a scale \(\mu\) enters, and the breaking shows up in experiment as \(1/137\to1/128\). The first place the tool breaks.
This document is Episode 11 of "c·t = const, That Clicks", written for physics-minded high-school and university readers. That adiabatic expansion gives \(n_\gamma\propto a^{-3}\), \(\rho_\gamma\propto a^{-4}\), \(T\propto a^{-1}\) and \(\lambda\propto a\); that their conformal weights are \(+3,-4,-1,+1\); and that the Maxwell action picks up \(\Omega^{D-4}\) and is conformally invariant only at \(D=4\), are all standard results (Episode 7 of the previous series). Today's CMB values \(T_0=2.7255\) K, \(n_\gamma=4.11\times10^8\ \mathrm{m^{-3}}\), \(s_\gamma=1.478\times10^9\,k_B\,\mathrm{m^{-3}}\) and \(\eta=6.1\times10^{-10}\) are standard; \(s_\gamma/n_\gamma=3.60\,k_B\) is computed here (exact blackbody value \(4\pi^4/45\zeta(3)=3.6017\)). During epochs of entropy release by annihilation there are corrections from the change in \(g_{*s}\) (Episode 2 computed 1.10 nat). "Nucleosynthesis is a function of \(\eta\) alone" is a simplification: \(g_*\), the neutron lifetime and the expansion rate also matter — and it was the expansion rate that ruled out \(a\propto t\) in Extra 1 of the previous series. The claim made here is only that swapping pictures does not move \(\eta\), so the picture cannot change the verdict. Taking \(A_\mu\) (lower index) to have weight 0 is the standard convention; the intermediate counting depends on the convention but the invariance of the action does not. That quantisation breaks conformal symmetry (the trace anomaly \(T^\mu{}_\mu=(\beta/2g)F_{\mu\nu}F^{\mu\nu}\)) and that \(\alpha^{-1}\) runs from 137.036 to 127.95 at \(M_Z\) are standard (Episode 8 of the previous series). The locally measured speed of light and the reading of a CMB thermometer are the same in every picture. 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).