It reads three ways ── one is dead, one is correct, and one is deep
Studying relativity, everyone thinks it at least once ── "the speed of light being constant is what forces spacetime to warp. So if the speed of light varies instead, can't space stay flat?"
And in particular \(ct=\) constant — measuring time so that light travels a fixed distance — looks as though it would simplify things.
This intuition contains something too good to throw away. Indeed one of the readings is completely correct and dramatically simplifies cosmological calculations. But another reading is a dead road the sister series already killed. Let us clearly separate which is which.
The single phrase "\(ct=\) constant" can mean at least one of the following three claims. Mix them and the argument is guaranteed to spin its wheels.
| reading | content of the claim | result |
|---|---|---|
| ① a varying speed of light | \(c\) changes with time (\(c\propto1/t\)) | dead |
| ② a choice of coordinates | choose coordinates in which light travels at \(45^\circ\) | correct. genuinely simplifies things |
| ③ scale invariance | demand that the universe has no intrinsic length | deep. reaches the core of the theory |
If \(ct=\) constant and \(t\) grows, then \(c\propto1/t\): the claim that light gradually slows. It is called a varying speed of light (VSL).
But it is not a physical claim. The reason is the sister series' spine, verbatim.
\(c\) carries dimensions, so "it changed" has no observer-independent meaning.
No experiment can even in principle distinguish "\(c\) halved" from "the definition of the metre doubled."
\(c\) is not a physical constant but a unit-conversion factor. Exactly the structure the sister series "Temperature That Clicks" identified for \(k_B\) ── just as \(k_B\) disappears if you measure temperature in energy, \(c\) disappears if you measure time in length. What can be eliminated cannot vary.
Only dimensionless quantities can vary. So to make VSL a physical claim, you have to say that
$$\alpha=\frac{e^2}{4\pi\varepsilon_0\hbar c}$$has changed. And the moment you say that, the story is not about "\(c\)" but about "\(\alpha\)". Not one line of the formulae gets simpler.
Now take \(ct=\) constant to mean choose coordinates in which light travels at \(45^\circ\). This is the standard and powerful tool called conformal time.
Write the metric of the expanding universe in these coordinates ──
$$ds^2=a(\eta)^2\left(-d\eta^2+d\vec x^{\,2}\right)$$Inside the bracket is just a flat metric. All information about the expansion has been pushed into the overall scale factor \(a(\eta)\).
And in four dimensions Maxwell's equations are conformally invariant. That is, the electromagnetic field does not feel an overall scale factor. Which means ──
In these coordinates, the expansion of the universe drops out of the equations as far as light is concerned.
A photon behaves as if in flat spacetime. The expansion is quarantined inside \(a(\eta)\) and does not appear in the electromagnetic equations of motion.
In the figure below, see the same light path in two coordinate systems.
On the left the light path bends. Because of the expansion, the comoving distance coverable in a given time keeps shrinking. Move to the right and ── it becomes a straight \(45^\circ\) line. The expansion has been absorbed into the coordinates and has vanished from the motion of light.
So far, good news. The universe could be written as "a flat metric × one scalar." So how far can this go?
The form "flat spacetime × a scalar" is called conformally flat, and there is an exact criterion.
the metric is conformally flat \(\iff\) the Weyl tensor vanishes
That draws the boundary sharply.
| spacetime | Weyl tensor | writable with a scalar alone? |
|---|---|---|
| FRW (expanding universe) | \(=0\) | yes. and exactly |
| Schwarzschild (a star) | \(\ne0\) | impossible in principle |
Section 03 working was no accident. FRW really is conformally flat. As far as cosmology goes, "space is not warped" is correct as a theorem.
A star is different. Being a vacuum solution, \(R_{\mu\nu}=0\), so all of its curvature is Weyl curvature.
this is the tidal force itself
Ricci is zero yet the curvature invariant is not. What remains is all Weyl. So it cannot be made conformally flat ── however you choose the scalar \(\Omega(x)\), you cannot write down a single star.
Posit the refractive-index picture \(n(x)\approx1-2\Phi/c^2\) and light bending, gravitational redshift and the Shapiro delay all come out correctly. It should not be writable ── so why?
The answer is one line. Because null geodesics are conformally invariant.
The path of light is determined by the conformal class of the metric alone and does not change when multiplied by \(\Omega^2\). So a picture that merely tunes the conformal factor is built so as to hit light structurally, without fail. And it does not hit anything but light ── not Weyl curvature, not the rate of clocks, not the orbits of matter.
The third reading takes \(ct=\) constant not as a constraint imposed but as a symmetry: that \(x^\mu\to\lambda x^\mu\) may be done freely. That is, it demands "the universe has no intrinsic scale."
Follow that direction all the way and you find something striking ── of the entire Lagrangian of known physics, only two terms carry dimensions: the Higgs mass term and Newton's constant. Delete those two and the whole theory becomes scale invariant.
For details see §05–06 of Bonus ①. Here let us note only what happens beyond that.
Finally, let us draw the most important line.
\(ct=\) constant is a choice of coordinates. And curvature is a tensor, so if it is non-zero in one coordinate system it is non-zero in all of them. "Choosing coordinates in which space does not warp" is impossible in principle.
The figure in §03 shows exactly this. Move to conformal time and light became exactly 45° ── the coordinate description became maximally simple. But \(a(\eta)\) is still sitting there.
The curvature did not vanish; it merely moved into the conformal factor.
Coordinates may be chosen freely. But invariants do not depend on the choice.
So is the position "space is not warped" completely wrong? No. A scalar was simply not enough, and supplying what was missing lets you build the whole of general relativity on a flat stage. That is Episode 3 of the main series.
① Read as "\(c\) varies," it dies. \(c\) carries dimensions, so "it changed" has no observer-independent meaning. It is a unit-conversion factor, not a physical constant. Only dimensionless quantities like \(\alpha\) can vary.
② Read as coordinates, it genuinely simplifies things. In conformal time the expanding-universe metric becomes "flat × a scale factor," and since Maxwell's equations are conformally invariant, the expansion vanishes as far as light is concerned. That light travels exactly \(45^\circ\) follows from \(ds^2=0\) and does not depend on the form of \(a(\eta)\).
③ The wall has a name ── the Weyl tensor. FRW has \(C=0\) and can be written exactly, but a star has \(C\ne0\) and cannot be written in principle. That light alone still comes out right is because null geodesics are conformally invariant. A hit from something bound to hit carries zero information.
④ Read as a symmetry, it reaches the core of the theory. Only two terms carry dimensions: the Higgs mass and Newton's constant. And at high-energy fixed points, scale does in fact lose its meaning.
And coordinates cannot erase curvature. Because curvature is a tensor. Make light 45° in conformal time and \(a(\eta)\) remains ── the curvature did not vanish, it merely moved.
This document is Bonus ② of the "Lattice We Build" series, a reading piece for high-school and university students who love physics. Where the sister series "That Clicks" explains known physics, this series shows the work itself.
Established material: that a change in a dimensionful constant has no observer-independent meaning; the conformal flatness of the FRW metric; the conformal invariance of Maxwell's equations in four dimensions; the conformal invariance of null geodesics; the equivalence of conformal flatness and the vanishing of the Weyl tensor; and the Kretschmann invariant of Schwarzschild. On the other hand, the scale-invariance picture of §05 (that all scales are generated from quantum anomalies and spontaneous breaking) is a position under research, not an established conclusion. And because of the conformal anomaly, exact scale invariance is generally broken as a quantum theory. The figure is schematic: the real universe is not described by a single power \(a\propto t^p\) (\(p\) shifts across radiation, matter and dark-energy domination).
Main series: Episode 1｜Episode 2｜Episode 3 ｜ Bonus ①: Could physics be written more simply? ｜ sister series: Relativity That Clicks／Temperature That Clicks ── to print, use your browser's "Print" → "Save as PDF."
Print / PDF: ⌘+P (Ctrl+P on Windows). The slider in the figure lets you watch the expansion being absorbed into the coordinates.