c·t = CONST, THAT CLICKS EPISODE 31 ／ Part IV — when the tool sits at the centre of a theory
CCC's central move is the one we counted in Episode 11
Penrose's conformal
cyclic cosmology
Gluing the end of the universe to the beginning of the next by a conformal transformation.
What happens when "mere notation" is placed at a theory's centre?
The second half of Part IV takes up theories that put the conformal transformation at their foundation. First, Penrose's conformal cyclic cosmology (CCC) — gluing the end of the universe to the beginning of the next by a conformal transformation. What happens when the operation this series has called "mere notation" is placed at the centre of a theory? And the move at CCC's centre is precisely the one we counted in Episode 11.
01At the centre is Episode 11's result
Episode 11 transformed every quantity of a photon gas — number density, energy density, temperature, wavelength. All invariant, because the power of \(a\) in the standard picture and the quantity's weight were the same number.
Episode 11's conclusion, restated in one line
With no mass there is no ruler; with no ruler the conformal factor has no meaning.
Penrose starts here — a far future in which all mass has gone, and a big bang consisting only of radiation, both have no "size". So why not glue them with a conformal transformation? That is CCC's central move.
Episode 11 wrote that light carries neither ruler nor clock (infinite Compton wavelength, zero proper time). CCC is that fact applied to the universe as a whole.
02Three conditions the gluing requires
All three can be measured with quantities this series has already counted. In order.
03(a) How much mass must be removed
all of which must lose its rest mass
| Process | Timescale | Basis |
|---|---|---|
| Proton decay | \(>2.4\times10^{34}\) yr | Super-Kamiokande lower bound |
| Stellar-mass BH evaporation (10 M☉) | \(2\times10^{70}\) yr | \(t=2.1\times10^{67}(M/M_\odot)^3\) yr |
| Galactic centre BH (\(10^9\) M☉) | \(2\times10^{94}\) yr | as above |
| Largest BHs (\(10^{11}\) M☉) | \(2\times10^{100}\) yr | as above — this is the gluing time |
04Where on the logarithmic axis does the gluing fall?
Conclusion of §04
In CCC's story the universe has made 140 of 348 moves —
only 40% of the way through its logarithmic run.
Episode 2 called the universe "a 140-move program"; in CCC it is still the first half.
Figure: CCC's story in logarithmic steps. Events on top, occupancy below (Episode 6). Move the slider to read what remains at each epoch — today sits 40% of the way from the left.
05(b) How empty does it get — measured by occupancy
what remains inside it (taking the Local Group to become one black hole)
$$M=10^{12}M_\odot\ \Longrightarrow\ \frac{S}{k_B}=\frac{4\pi GM^2}{\hbar c}=1.05\times10^{101}$$| Occupancy | From | |
|---|---|---|
| Today | \(1.5\times10^{-18}\) | Episode 6 |
| Far future (de Sitter) | \(3.2\times10^{-22}\) | today — 3.7 orders emptier still |
| The moment of gluing | \(0\) | CCC's requirement (\(C=0\)) |
Episode 6 counted that the universe began thermally full and gravitationally empty and is now emptying. That direction continues all the way to the end — and CCC connects to the next universe at that endpoint (completely empty). Episode 6's "how badly the tool is breaking is the arrow of time" inverts in CCC into "when the tool works perfectly again, the universe ends".
06(c) Cancelling the entropy — where the bet is placed
The third condition is the heaviest. \(S/k_B=3.1\times10^{104}\) has already accumulated, and the next universe must begin with low entropy. Where does it go?
Penrose's answer
Information is lost in black hole evaporation.
The phase space itself shrinks, so the ceiling on entropy comes down.
This places a clear bet on one side of the black hole information problem. The mainstream position, grounded in AdS/CFT, is that information is not lost (unitarity); CCC does not take it. This document does not adjudicate that debate, but it does make explicit where CCC has bet.
07An observational claim — Hawking points
This is the important part about CCC: it carries an observable prediction.
| Claim | Content |
|---|---|
| Hawking points | black hole evaporation in the previous aeon leaves circular marks in the CMB |
| Reports | Gurzadyan & Penrose (2010, 2013), Penrose et al. (2018) |
| Rebuttals | Wehus & Eriksen (2011), DeAbreu et al. (2015), Jow & Scott (2020) |
| Substance of the rebuttals | the same statistics appear in Gaussian \(\Lambda\)CDM simulations |
Unsettled. But carrying an observable claim at all matters — Episode 28's phase-transition VSL escaped exclusion and lost every prediction in the observable era at the same stroke. CCC has not done that.
08The surgery — CCC's name matches its content
| Ep. | Theory | (A) notation | (B) observable claim | Result of the surgery |
|---|---|---|---|---|
| 27 | Inflation | establishing causal contact | \(n_s\) | (B) survives |
| 28 | VSL | a change of units | \(\alpha\) varies | chose (B), kept (A)'s name |
| 29 | MOND | positing \(a_0\) | dynamics set by \(g/a_0\) | (B) is the substance |
| 31 | CCC | the conformal gluing | the previous aeon continues (Hawking points) | uses (A) openly as a tool and places its claim in (B) |
The thing this episode most wants to say
CCC is the theory in Part IV that best withstands the surgery.
Its name (conformal cyclic cosmology) correctly points at both (A) and (B) —
it uses the conformal transformation as a tool, knowing it is "mere notation".
Episode 28's VSL claimed (B) while keeping (A)'s name, and the 26 bits of constraint on \(\alpha\) stopped being visible. CCC does the reverse: it states explicitly that at the moment of gluing there is no ruler, so the conformal factor is meaningless, and places (B) on top of that. The surgery this series has used since Episode 3 has already been performed by the theory itself.
① CCC is a minority hypothesis and is not widely accepted as an alternative to standard cosmology. This document neither supports nor refutes it, and measures only what this series' tools (conformal weights, occupancy, logarithmic steps) can measure.
② The mechanism by which the electron loses its rest mass is unverified. Charge conservation makes the electron stable as the lightest charged particle, with no known decay channel — a known weak point of CCC, stated by Penrose as a conjecture. Condition (a) stops here.
③ §04's "348 moves" uses \(t=2.1\times10^{67}(M/M_\odot)^3\) yr with \(M=10^{11}M_\odot\). The actual gluing time depends on the mass of the heaviest black hole in the universe and moves by tens of steps. Read "today is at 40%" with the same precision.
④ §05's \(3.2\times10^{-22}\) assumes crudely that the Local Group (\(10^{12}M_\odot\)) becomes one black hole. How much mass actually remains inside the horizon depends on the nature of dark energy and the dynamics of the Local Group — an order-of-magnitude argument.
⑤ The Hawking point claim is disputed. This document does not adjudicate. The rebuttals (the same statistics appear in Gaussian \(\Lambda\)CDM simulations) come from several independent groups, and there is no positive consensus at present.
⑥ §06's "betting on information loss" summarises CCC's position. The black hole information problem is unresolved and this document does not judge it, including the unitarity side — it only makes explicit which way CCC has bet.
Exercises (solvable with this episode's formulas alone)
- State CCC's central move in Episode 11's language.
Show the answer
Episode 11 counted that every quantity of a photon gas is conformally invariant — that is, with no mass there is no ruler, and with no ruler the conformal factor has no meaning. So "a far future with all mass gone" and "a big bang of radiation only" both have no size, and can be glued by a conformal transformation. - Name the three conditions the gluing requires.
Show the answer
(a) all rest mass disappears, (b) the Weyl curvature \(C\) goes to zero (the Weyl curvature hypothesis), (c) entropy is cancelled. All three can be measured with quantities this series has already counted. - Where does the gluing fall in logarithmic steps, and what fraction is today?
Show the answer
The largest BHs (\(10^{11}M_\odot\)) evaporate in \(2\times10^{100}\) yr, so \(\ln(t/t_P)=347.9\). Today is 140.2, i.e. 40%. Episode 2's "140 moves" is, in CCC's story, still the first half. - Find the far-future occupancy.
Show the answer
The de Sitter horizon gives \(S_{\rm dS}/k_B=3.31\times10^{122}\), and the remaining Local Group black hole \(1.05\times10^{101}\). The ratio is \(3.2\times10^{-22}\) — 3.7 orders emptier than today's \(1.5\times10^{-18}\). Episode 6's emptying continues to the end. - (Harder) Why must CCC bet on information loss?
Show the answer
Episode 16 counted that "a conformal transformation can move only the unused side", and Episode 6 that "the memory in use today is almost entirely black holes — the Weyl side". Together — the conformal transformation used for the gluing cannot reach that \(3.1\times10^{104}\). So condition (c) cannot be explained conformally and needs another mechanism. CCC's bet on information loss is a consequence of this structure.
Summary — when the tool is placed at the centre of a theory
The move at CCC's centre is exactly the one we counted in Episode 11 — every quantity of a photon gas is conformally invariant, so with no mass there is no ruler, and with no ruler the conformal factor has no meaning. Hence "a far future with all mass gone" and "a big bang of radiation only" can be glued.
We measured the three conditions with this series' quantities. (a) The rest mass to be removed is 31.4% of today's energy, and the largest black holes evaporate in \(2\times10^{100}\) yr — step 348 on the logarithmic axis, with today still at 40%. (b) Occupancy falls from today's \(1.5\times10^{-18}\) to \(3.2\times10^{-22}\), 3.7 orders emptier — Episode 6's direction continuing to the end.
(c) is where the bet sits. Where does the \(3.1\times10^{104}\) already accumulated go? Penrose's answer is information loss in black hole evaporation, a clear side of the black hole information problem. And this series' tools show why he has no choice — Episode 16 ("a conformal transformation moves only the unused side") plus Episode 6 ("the memory in use is almost entirely the Weyl side") equals the gluing transformation cannot reach that \(3.1\times10^{104}\).
And the surgery — CCC withstands it best of anything in Part IV. Episode 28's VSL failed by claiming (B) while keeping (A)'s name; CCC states explicitly that at the moment of gluing there is no ruler and hence no meaningful conformal factor, and places its claim in (B) — Hawking points. The surgery this series has used since Episode 3 was already performed by the theory itself. The Hawking point claim is unsettled, but carrying an observable claim at all is the decisive difference from phase-transition VSL.
This document is Episode 31 of "c·t = const, That Clicks", written for physics-minded high-school and university readers. Conformal cyclic cosmology is due to Penrose (2010, Cycles of Time). The conformal invariance of massless fields, the Weyl curvature hypothesis, the black hole evaporation time \(t\simeq2.1\times10^{67}(M/M_\odot)^3\) yr, and the Bekenstein–Hawking entropy \(S=4\pi GM^2/\hbar c\,k_B\) are all standard, as is the Super-Kamiokande proton lifetime bound \(>2.4\times10^{34}\) yr. The logarithmic steps (today 140.2, gluing 347.9, today at 40% of the run), the far-future occupancy \(3.2\times10^{-22}\), and the 31.4% of rest mass to be removed are computed here (kenshou/calc35.py). CCC is a minority hypothesis and is not widely accepted as an alternative to standard cosmology — this document neither supports nor refutes it and measures only what this series' tools can measure. The mechanism by which the electron loses its rest mass is unverified, and this is a known weak point of CCC (Penrose states it as a conjecture). §04's 348 steps depend on the mass of the heaviest black hole and move by tens of steps. §05's \(3.2\times10^{-22}\) assumes crudely that the Local Group (\(10^{12}M_\odot\)) becomes one black hole — an order-of-magnitude argument. The Hawking point claim is disputed: against the reports of Gurzadyan & Penrose (2010, 2013) and Penrose et al. (2018), Wehus & Eriksen (2011), DeAbreu et al. (2015) and Jow & Scott (2020) argue that the same statistics appear in Gaussian \(\Lambda\)CDM simulations, and there is no positive consensus at present. §06's "betting on information loss" summarises CCC's position; the black hole information problem is unresolved and is not judged here. Linear expansion (\(c\cdot t=\)const, \(R_h=ct\)) is a minority model under examination. 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).