c·t = CONST, THAT CLICKS EPISODE 33 ／ Part IV — the two hardest cases to tell apart
There is one way to tell them apart — look at \(k\)
Milne versus
\(R_h=ct\)
Both have \(a\propto t\). Yet Milne returns to flat spacetime by a change of coordinates and \(R_h=ct\) does not.
Coordinate change, or conformal transformation? We build the procedure.
Part IV's two most easily confused cases, side by side — the Milne universe and \(R_h=ct\). Both have "\(a\propto t\)" and both are described as "neither decelerating nor accelerating". And they are entirely different things. Episode 3 said "\(c\cdot t=\)const is not a coordinate change but a conformal transformation"; here we apply that distinction to the hardest case to tell apart, and build a procedure for telling them apart.
01Both have \(a\propto t\); only \(k\) and the contents differ
| Milne universe | \(R_h=ct\) | |
|---|---|---|
| Scale factor | \(a\propto t\) | \(a\propto t\) |
| Contents | empty (\(\rho=0\)) | contains matter |
| Spatial curvature | \(k=-1\) | \(k=0\) |
| Total equation of state | ── | \(w=-1/3\) |
The visible \(a(t)\) is identical. Only \(k\) and the contents differ — and that turned out to be decisive.
02Compute the scalar curvature and it is settled
the scalar curvature of FLRW (\(c=1\))
$$R=6\left[\frac{\ddot a}{a}+\left(\frac{\dot a}{a}\right)^2+\frac{k}{a^2}\right]$$with \(a=t\), so \(\ddot a=0\) and \(\dot a/a=1/t\)
$$R=\frac{6(1+k)}{t^2}$$| \(k\) | \(R\) | What it is |
|---|---|---|
| \(-1\) | \(0\) | Milne — exactly flat |
| \(0\) | \(6/t^2\) | \(R_h=ct\) |
| \(+1\) | \(12/t^2\) | (for reference: the closed case) |
Conclusion of §02
Exactly zero at \(k=-1\). And not only the scalar curvature — the entire Riemann tensor vanishes identically.
── The Milne universe is Minkowski spacetime in different coordinates.
03The heart — a three-step test
| Spacetime | Riemann | Weyl \(C\) | What is required |
|---|---|---|---|
| Minkowski | \(0\) | \(0\) | nothing at all |
| Milne | \(0\) | \(0\) | coordinates only (Step 1) |
| \(R_h=ct\) | \(\ne0\) | \(0\) | a conformal transformation (Step 2) |
| \(\Lambda\)CDM | \(\ne0\) | \(0\) | a conformal transformation (Step 2) |
| Schwarzschild | \(\ne0\) | \(\ne0\) | neither can remove it (Step 3) |
The thing this episode most wants to say
Every FLRW falls in Step 2 (Weyl zero, Ricci non-zero).
── The place this series has occupied since Episode 1 is pinned down by that one line.
04Compare luminosity distances and something surprising happens
| \(z\) | \(\Lambda\)CDM | Milne | \(R_h=ct\) | \(\Delta\mu\) Milne | \(\Delta\mu\) \(R_h=ct\) |
|---|---|---|---|---|---|
| 0.3 | 0.3613 | 0.3450 | 0.3411 | −0.101 | −0.125 |
| 0.5 | 0.6580 | 0.6250 | 0.6082 | −0.112 | −0.171 |
| 1.0 | 1.5292 | 1.5000 | 1.3863 | −0.042 | −0.213 |
| 1.5 | 2.5188 | 2.6250 | 2.2907 | +0.090 | −0.206 |
| 2.0 | 3.6837 | 4.0000 | 3.2958 | +0.179 | −0.181 |
Conclusion of §04
Remarkably, at \(z=1\) Milne is closer to \(\Lambda\)CDM (1.500 against 1.529, a gap of 0.042 mag).
An empty spacetime fits the Hubble diagram better than \(R_h=ct\) does.
── "Fits the Hubble diagram" is a weak test of a model.
Figure: departure from \(\Lambda\)CDM in magnitudes. The grey band is one supernova's intrinsic scatter (0.15 mag). Move the slider through redshift to read how many supernovae are needed for a 5σ distinction at that \(z\).
05And yet Milne is decisively excluded
| Observation | Why it fails |
|---|---|
| CMB acoustic peaks | no baryons, so no sound waves |
| Nucleosynthesis | no matter, so no \(^4\)He and no D |
| Structure formation | no matter to gather gravitationally |
| Measurement of \(\Omega_m\) | \(\Omega_m=0.315\pm0.007\) — more than 45σ from \(\Omega_m=0\) |
Good geometry with nothing in it is not physics. In Episode 25's terms — \(L(\text{residual})\) is not determined by the Hubble diagram alone. On that one dataset Milne looks preferable, and the moment other datasets enter it evaporates (the same structure as MOND in Episode 29).
06The reveal — the source of the confusion, and how to resolve it
| Milne | \(R_h=ct\) | |
|---|---|---|
| How it is described | "\(a\propto t\), neither decelerating nor accelerating" | the same |
| \(k\) | \(-1\) | \(0\) |
| Riemann | \(0\) | \(\ne0\) |
| Operation required | a coordinate change | a conformal transformation |
| Physical content | zero | masses move |
Conclusion of §06
There is one way to tell them apart — look at \(k\).
\(k=-1\) (and empty) means coordinates suffice and the physical content is zero.
\(k=0\) (with matter) means a conformal transformation is required and masses move.
── This is the most concrete meaning of Episode 3's "not a coordinate change but a conformal transformation".
07Where has this series been?
The range where this series' tool works coincides exactly with the Step 2 row. Episode 13 measured "a conformal transformation touches only size" and Episode 16 "it can move only the unused side" — today restates those in the language of geometry.
① That the Milne universe is flat spacetime is an exact result. The coordinate change \(T=t\cosh\chi,\ R=ct\sinh\chi\) makes it a portion of Minkowski spacetime (inside the light cone) — "a portion" matters: Milne coordinates do not cover all of Minkowski spacetime.
② "Every FLRW has Weyl \(=0\)" follows from homogeneity and isotropy. Add perturbations and \(C\ne0\), moving it to Step 3 — the real universe is not exactly FLRW, which is why Episode 6's occupancy is not zero (\(1.5\times10^{-18}\)). The tables in §02 and §03 are statements about the background spacetime.
③ §04's \(\Lambda\)CDM comes from a numerical integration here with \(\Omega_m=0.315\), \(\Omega_r=9.2\times10^{-5}\). Milne's \(H_0d_L/c=z(1+z/2)\) and \(R_h=ct\)'s \((1+z)\ln(1+z)\) are closed forms. Real supernova analyses leave the absolute magnitude (a constant offset) free, so the \(\Delta\mu\) here cannot be converted directly into significance — the figure's "supernovae needed" is an indication for a fixed constant offset.
④ "Milne is closer to \(\Lambda\)CDM" holds near \(z\simeq1\). At \(z=0.5\) it is not much different from \(R_h=ct\), and at \(z=2\) Milne is the further one (\(+0.179\) mag). Picking a particular \(z\) changes the story, so read it as an order and a trend.
⑤ The verdict on \(R_h=ct\) was handled in Episode 3. This document does not re-examine it and treats only its structural difference from Milne. The academic standard remains \(\Lambda\)CDM.
Exercises (solvable with this episode's formulas alone)
- Find the scalar curvature of an FLRW with \(a=t\) and show it vanishes at \(k=-1\).
Show the answer
Into \(R=6[\ddot a/a+(\dot a/a)^2+k/a^2]\) put \(a=t\): \(\ddot a=0\), \(\dot a/a=1/t\), \(k/a^2=k/t^2\), so \(R=6(1+k)/t^2\). Exactly zero at \(k=-1\). - What is the essential difference between Milne and \(R_h=ct\)?
Show the answer
Whether the Riemann tensor vanishes. Milne's vanishes identically (it is Minkowski in other coordinates); \(R_h=ct\) has \(R=6/(ct)^2\ne0\). So Milne needs only a coordinate change, and \(R_h=ct\) needs a conformal transformation. - State the three-step test and classify \(\Lambda\)CDM and Schwarzschild.
Show the answer
Step 1: Riemann \(=0\) → coordinates suffice. Step 2: Riemann \(\ne0\) but Weyl \(=0\) → a conformal transformation is required. Step 3: Weyl \(\ne0\) → neither works. \(\Lambda\)CDM is Step 2 (as is every FLRW); Schwarzschild is Step 3. - At \(z=1\), which is closer to \(\Lambda\)CDM?
Show the answer
Milne (1.500 against \(\Lambda\)CDM's 1.529, a gap of 0.042 mag); \(R_h=ct\) gives 1.386, a gap of 0.213. An empty spacetime fits the Hubble diagram better than \(R_h=ct\) — which is why "fits the Hubble diagram" is a weak test. - (Harder) State the range where this series' tool works, in the language of the three-step test.
Show the answer
It coincides exactly with the Step 2 row. At Step 1 (Milne) the spacetime is already flat and not even the masses need to move; at Step 3 (Schwarzschild) a conformal transformation cannot flatten it. Every FLRW being at Step 2 is precisely what allowed Episode 4's "delete everything deletable and be left with one mass" — a geometric restatement of the tool's limit as measured in Episodes 13 and 16.
Summary — look at \(k\) and it is settled
The Milne universe and \(R_h=ct\) both have \(a\propto t\) and are both described as "neither decelerating nor accelerating". Only \(k\) and the contents differ — and computing the scalar curvature settles it. With \(a=t\), \(R=6(1+k)/t^2\), which is exactly zero at \(k=-1\). The whole Riemann tensor vanishes identically, so Milne is Minkowski spacetime in different coordinates. \(R_h=ct\) has \(R=3.5\times10^{-52}\ \mathrm{m^{-2}}\) (curvature radius 1.73 Gpc), which is not zero.
From this a general three-step test follows — Step 1: Riemann \(=0\), coordinates suffice and there is no physical content. Step 2: Weyl \(=0\), a conformal transformation is required and masses move. Step 3: Weyl \(\ne0\), neither works. Every FLRW falls in Step 2 — the place this series has occupied since Episode 1, pinned down by one line.
Comparing luminosity distances gives a surprise. At \(z=1\), Milne is closer to \(\Lambda\)CDM (0.042 mag against \(R_h=ct\)'s 0.213) — an empty spacetime fits the Hubble diagram better. And yet Milne is decisively excluded by the CMB, nucleosynthesis, structure formation and the measurement of \(\Omega_m\) (over 45σ). Good geometry with nothing in it is not physics — the same structure as Episode 25's "\(L(\text{residual})\) is not the Hubble diagram alone" and Episode 29's "the question is not one question".
And the reveal — tell them apart by looking at \(k\). \(k=-1\) (and empty) means coordinates suffice and the physical content is zero; \(k=0\) (with matter) means a conformal transformation is required and masses move. This is the most concrete meaning of Episode 3's "not a coordinate change but a conformal transformation".
This document is Episode 33 of "c·t = const, That Clicks", written for physics-minded high-school and university readers. That the Milne universe (Milne 1935) is an empty open FLRW (\(\rho=0\), \(k=-1\), \(a\propto t\)) and becomes a portion of Minkowski spacetime under the coordinate change \(T=t\cosh\chi,\ R=ct\sinh\chi\) is a standard result — "a portion" matters: Milne coordinates do not cover all of Minkowski spacetime. The FLRW scalar curvature \(R=6[\ddot a/a+(\dot a/a)^2+k/a^2]\) and the conformal flatness of FLRW (vanishing Weyl tensor) are also standard. The relation \(R=6(1+k)/t^2\), today's value \(3.52\times10^{-52}\ \mathrm{m^{-2}}\) for \(R_h=ct\) (curvature radius 1.73 Gpc), and the luminosity distance comparison are computed here (kenshou/calc37.py). "Every FLRW has Weyl \(=0\)" is a statement about the homogeneous, isotropic background; perturbations give \(C\ne0\) — the real universe is not exactly FLRW, which is why Episode 6's occupancy is not zero. The \(\Lambda\)CDM values come from a numerical integration here with \(\Omega_m=0.315\), \(\Omega_r=9.2\times10^{-5}\); Milne's \(H_0d_L/c=z(1+z/2)\) and \(R_h=ct\)'s \((1+z)\ln(1+z)\) are closed forms. Real supernova analyses leave the absolute magnitude free, so the \(\Delta\mu\) here cannot be converted directly into significance — the figure's "SNe for 5σ" assumes a fixed constant offset. "Milne is closer to \(\Lambda\)CDM" holds near \(z\simeq1\) and reverses by \(z=2\). \(\Omega_m=0.315\pm0.007\) is the Planck value. The verdict on \(R_h=ct\) was handled in Episode 3; this document treats only its structural difference from Milne. Linear expansion (\(c\cdot t=\)const, \(R_h=ct\)) is a minority model under examination, and 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).