Light has no ruler — so as far as light is concerned, a conformal transformation is nothing at all
In Episode 5, the only matter that left conformal symmetry alone was radiation (\(T^\mu{}_\mu=(3w-1)\rho c^2\) vanishes only at \(w=1/3\)). In Episode 6, the dimensionless ratio that refused to vanish contained \(m\). The culprit is the same in both cases — mass. Today we see, in two lines of algebra, why light alone can pass straight through a conformal transformation. And then we meet someone who tried, on the strength of that one point, to stitch the end of the universe to its beginning.
Write down the scoring table of the electromagnetic field (the Maxwell action) — in \(D\) dimensions, for generality.
$$S_{\rm EM}=-\frac{1}{4\mu_0}\int\!\sqrt{-g}\;F_{\mu\nu}F_{\alpha\beta}\,g^{\mu\alpha}g^{\nu\beta}\;d^Dx$$Now apply the conformal transformation \(g_{\mu\nu}\to\Omega^2g_{\mu\nu}\). Only two places are affected.
① The volume element gives birth to \(D\) factors of \(\Omega\)
$$\sqrt{-g}\;\longrightarrow\;\Omega^{D}\sqrt{-g}$$② Two inverse metrics eat 4 of them
$$g^{\mu\alpha}g^{\nu\beta}\;\longrightarrow\;\Omega^{-4}g^{\mu\alpha}g^{\nu\beta}$$\(A_\mu\) keeps its index downstairs and picks up nothing (weight 0), and \(F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu\) is likewise untouched. So what is left over is
$$S_{\rm EM}\;\longrightarrow\;\Omega^{\,D-4}\times S_{\rm EM}$$The number born and the number eaten balance only when \(D=4\).
| Dimension \(D\) | Left-over factor | Result |
|---|---|---|
| 2 | \(\Omega^{-2}\) | not conformally invariant |
| 3 | \(\Omega^{-1}\) | not conformally invariant |
| 4 | \(\Omega^{0}=1\) | exactly conformally invariant |
| 5 | \(\Omega^{+1}\) | not conformally invariant |
| 6 | \(\Omega^{+2}\) | not conformally invariant |
We met the specialness of four dimensions in Episode 4 as well — the conformal coupling \(\xi=(D-2)/4(D-1)\) equalling \(1/6\), at that same \(D=4\). The dimension we live in gets along suspiciously well with conformal transformations. Both light and gravity ride cleanly into the conformal language only in four dimensions.
Why, then, does mass snag? As we saw in Episode 3, under a conformal transformation mass must move as \(\tilde m=\Omega^{-1}m\) or the mass term fails to keep its form. In other words you cannot hold the mass fixed while changing only the ruler.
The reason takes one line.
Fix a mass \(m\) and a definite length is fixed along with it. A definite length means definite marks on a ruler. A conformal transformation is the operation "you may swap the ruler freely", so of course it collides with anything that insists on its own marks.
Light has no such length. As \(m\to0\), \(\lambda_C\to\infty\) — it specifies no length at all. So light has nothing to say about a conformal transformation. Tell it "I swapped the ruler" and it has nothing to compare against, so it cannot notice.
Let us now pin down, in the deepest way available, what a conformal transformation does.
When \(g_{\mu\nu}\) becomes \(\Omega^2g_{\mu\nu}\), the vectors \(v\) satisfying \(g(v,v)=0\) do not change (because \(\Omega^2\cdot0=0\)). That is, light cones do not move at all. A conformal transformation does not touch causal structure — what can influence what.
Count the components and the situation becomes clear.
A conformal transformation can move only that last one. Nine tenths of the information in spacetime is beyond its reach. So when we say "the transformation erased the expansion" (Episodes 3–6), what was erased is only that one part in ten — from which the survival of the dimensionless ratio in Episode 6 starts to look inevitable.
Drag the knob to the far left. The green circles swell out of the frame and disappear, and only the amber cones remain. That is what a massless world looks like — causal structure and nothing else, with no standard of size. In such a world you may stretch spacetime by \(\Omega\) and nobody can notice.
Now the real subject. A "world where mass does not matter" is not a thought experiment. The universe actually has two such epochs.
Both ends become worlds without a ruler. And so Penrose thought: if two worlds have no ruler, might they not be glued together freely by a conformal transformation?
The skeleton of Roger Penrose's conformal cyclic cosmology (CCC) is this.
Take the far future of one universe (Penrose calls it an "aeon"). Its end, after accelerated expansion has spread it out without limit, lies infinitely far away as geometry — but what is there is only radiation, which has no ruler. So a conformal transformation can shrink infinity down to something finite and draw it as a boundary. It is exactly the inverse of what we did in Episode 6 when we pushed the Big Bang out to \(\eta=-\infty\).
Meanwhile, just after the next universe's Big Bang, things are too hot for mass to count. That side too can be stretched from zero out to something finite by a conformal transformation and drawn as a boundary.
the infinite future of the previous aeon = the Big Bang of the next
at the seam there is no rest mass, \(T^\mu{}_\mu=0\), and the two spacetimes are joined by a conformal rescaling
To borrow Penrose's way of putting it — the "feel" of a universe that has expanded to infinity is indistinguishable from the "feel" of a newborn, highly dense one. If you have no means of measuring size, there is no large and no small.
What does the work here is only the fact we have been building since section 01: anything without mass passes straight through a conformal transformation. On that single point he tried to sew the end of the universe to its beginning. The audacity is admirable.
CCC is an unverified hypothesis. At least three large debts remain.
① Does rest mass really disappear? There is no evidence that the electron decays. Penrose assumes some decay of mass, but the Standard Model contains no such mechanism.
② There is no way to fix the conformal factor \(\Omega\) uniquely. No prescription for how to sew is settled, and different researchers' proposals disagree — this is CCC's most basic open point.
③ The observational evidence is not settled. Penrose and collaborators claim to see traces of a previous aeon in the CMB (concentric structures), but the debate over statistical significance continues.
This series' position is consistent: a conformal transformation is by itself a rewriting, not physics. For CCC to become physics it needs a genuinely new assumption like ①, and that assumption has not been paid for. Recall the counting in Episode 4 (one field, minus one symmetry, net zero). What comes for free is just bookkeeping.
In the Maxwell action the volume element gives birth to \(D\) factors of \(\Omega\) and two inverse metrics eat four, leaving \(\Omega^{D-4}\) — exactly conformally invariant in four dimensions and nowhere else. Mass, on the other hand, imports a definite length, the Compton wavelength \(\hbar/mc\), and therefore collides with the swapping of rulers. Light has no such length (not even a proper time) and knows nothing of scale. So it feels nothing.
Said more deeply: of the 10 components of a four-dimensional metric, a conformal transformation can move only 1 (the marks on the ruler); the remaining 9 (light cones = causal structure) are beyond its reach. And the universe has two epochs where mass stops mattering — a beginning too hot and an end too thin. Penrose tried to sew those two ends together with a conformal transformation (CCC). The idea rests on the single point established in this episode, but three things remain unresolved: ① whether rest mass really disappears, ② how to fix the conformal factor uniquely, and ③ the observational evidence.
Print / PDF: ⌘+P (Ctrl+P on Windows). On screen the slider changes the mass and shows the rulers vanishing while the light cones remain. "Show answer" opens the solutions.