"Which one is real?" — ten episodes spent building a shape in which that question can be answered
"Is it expanding, is light slowing down, or are atoms shrinking?" — when Episode 3 set the three pictures side by side, everyone must have thought the same thing. So which is it, really? This series has kept answering "any of them, they are gauge". But is that a dodge? In the finale we set out the two positions that take this question seriously, and then give the series' answer. To say it first — only dimensionless invariants are entitled to answer. And that is not a dodge.
The position represented by Flanagan is unambiguous: the effort to decide which frame is correct is beside the point, at least within classical physics. Why? Because every observable quantity is a conformal-frame invariant.
This is exactly what the series has been demonstrating. In Episode 3 we computed the redshift in three pictures and got \(1+z=t_0/t_e\) every time. The dimensionless ratio at recombination was 52.6 in either picture. In Episode 4 we saw "distance between galaxies ÷ atomic radius" is gauge independent. In Episode 6, \(N=mc^2t/\hbar\) came out as literally the same expression in two pictures. If everything measurable agrees, no experiment exists that decides which is real.
Observables = conformal-frame invariants.
Therefore the question "which frame is physical?" has no content.
Faraoni and others reply: once matter is included it is not so simple.
The reason is concrete. Under a conformal transformation, every quantity with mass dimension is transformed (Episode 3's table of weights). So a particle that was in free fall in the original frame — moving along a geodesic — does not follow a geodesic in the transformed frame. It feels a force from the dilaton's gradient.
The original frame
Mass is constant. The particle "goes straight through curved spacetime" — a geodesic.
The transformed frame
Spacetime is flat. But mass varies from place to place as \(\tilde m=\phi m\), so the particle feels a force from the gradient of mass and bends.
The same motion, seen either as "straight, on a curved floor" or as "on a flat floor, under a force". The trajectory is the same either way, but the definition of "free fall" differs between frames.
Hence Faraoni and others say: if you want to speak plainly about "free fall" or the equivalence principle, you are forced to declare one of the frames to be the physical one.
This is the most important point of the episode. A and B are not actually quarrelling. The whole dispute is about what you call an observable.
So much for the classical story. Quantise and it genuinely becomes unclear. Episodes 8 and 9 have supplied the reasons.
Episode 8's anomaly. Quantisation required a standard of resolution \(\mu\). But swapping frames transforms that standard too. The calculations in two frames may differ by anomaly terms — two descriptions that were exactly identical classically may cease to be identical quantum mechanically.
Episode 9's ghost. In a path integral, "which field you choose as the integration variable" is itself the choice of frame. And because of the conformal factor's reversed sign, that choice affects whether the integral converges.
The state of the literature is, honestly, a mess — there are several papers concluding "equivalent even at the quantum level" and several reporting "it breaks". This is an open problem.
Here is the answer we have spent ten episodes preparing.
Only dimensionless invariants are entitled to answer "which frame is real".
If one moves, it is physics; if it does not, it is bookkeeping.
And that verdict can always be delivered.
This is not a dodge, because it is a decision procedure. Whatever claim arrives, the same steps handle it. "The universe is not expanding, atoms are shrinking" (Episode 2)? Build the dimensionless ratio. "The singularity is a coordinate artefact" (Episode 6)? Build the dimensionless ratio. "\(\alpha\) is varying in time" (bonus episode 3 of the previous series)? Build the dimensionless ratio. The answer comes out mechanically, every time.
| Ep. | What we moved (bookkeeping) | What remained (physics) |
|---|---|---|
| 3 | The choice of time coordinate (\(t\leftrightarrow\eta\)) — this is what makes the speed of light change | \(1+z=t_0/t_e\), the recombination ratio 52.6, \(\alpha\) |
| 4 | The gauge \(s\) (splitting the expansion between metric and field) | \(\tilde a\cdot\phi\) = distance between galaxies ÷ atomic radius |
| 5 | The apparent shape of the potential (an effective potential) | The equation of state \(w\); the triple boundary at \(w=-1/3\) |
| 6 | Curvature \(R\) (it carries units; you can even set \(\tilde R=0\)) | \(N=mc^2t/\hbar\), the Planck time |
| 7 | The marks on the ruler (1 of the metric's 10 components) | Light cones = causal structure (the other 9) |
| 8 | ── | The \(\beta\) function, \(1/137\to1/128\), 99% of the proton's mass |
| 9 | ── | The conformal factor problem (no gauge can erase it) |
Only the right-hand column says anything about the universe. The left-hand column is about how we choose to describe it. What we spent ten episodes doing was separating these two columns accurately.
Let us be honest and list what has not closed.
The arrangement of positions A and B is this series' reading. The two camps have not shaken hands over "the dispute is about the definition of observables" — the actual debate is more involved, and opinions still differ in particular about how to handle matter with non-minimal coupling, and about how quantum corrections act.
And the conclusion in section 05 is not universal. "Build a dimensionless invariant" is a decision procedure; it does not tell you which dimensionless quantity to build. We chose \(N=mc^2t/\hbar\) in Episode 6 because we already knew it had physical meaning. The procedure delivers answers mechanically, but framing a good question is, as always, human work.
"Which frame is real" has two positions. A (Flanagan and others): all observables are conformal-frame invariants, so the question has no content. B (Faraoni and others): with matter included, geodesics do not carry over, so one frame must be declared physical. They do not conflict — the dispute is only about what counts as an observable. The numbers coming out of the laboratory agree (A); the words used to explain them do not (B). At the quantum level, with the anomaly and the ghost entangled, nothing is settled.
This series' answer is a decision procedure: only dimensionless invariants are entitled to answer. If it moves, physics; if not, bookkeeping. What ten episodes did was separate those two columns accurately. The choice of time coordinate, the gauge \(s\), the curvature \(R\), the marks on the ruler — all left column. \(1+z\), \(N=mc^2t/\hbar\), light cones, the \(\beta\) function — all right column. And Episode 9's conformal factor problem alone cannot be moved to the left by any choice of gauge — there really are things that rewriting the books cannot erase.
Print / PDF: ⌘+P (Ctrl+P on Windows). On screen the slider changes the gauge and shows that the four on the right do not move. "Show answer" opens the solutions.