c·t = CONST, THAT CLICKS EPISODE 38 ／ Part V — where the tool breaks
The ghost never disappears; it only moves
A kinetic term
with the wrong sign
Pull the conformal factor out of the Einstein action and its kinetic term alone flips sign.
The Euclidean path integral has no bottom.
Last time the breaking came from quantum theory being unable to stay at \(D=4\). This time, something worse — pull the conformal factor out of the Einstein action and its kinetic term alone comes out with the wrong sign. The energy has no lower bound and the Euclidean path integral diverges. This is the source of the ghost we met twice in Episode 34. And what it shows is this — the ghost never disappears; it only moves.
01Pull out the conformal factor and a kinetic term appears
after integrating by parts, the coefficient of \(\Omega\)'s kinetic term is \((D-1)(D-2)\) — with the sign opposite to an ordinary scalar field
| \(D\) | \((D-1)(D-2)\) | meaning |
|---|---|---|
| 1 | \(0\) | exactly zero |
| 2 | \(0\) | exactly zero — in two dimensions the conformal factor has no kinetic term |
| 3 | \(2\) | |
| 4 | \(6\) | our dimension |
| 5 | \(12\) | |
| 6 | \(20\) | |
| 10 | \(72\) |
Conclusion of §01
At \(D=4\) the coefficient is 6. Only \(D=1\) and \(D=2\) make it vanish.
── In our dimension there is no way out.
02The conformal factor is a conformally coupled scalar with the wrong sign
| \(D\) | \(\xi\) | |
|---|---|---|
| 3 | \(0.1250\) | |
| 4 | \(0.1667\) | our dimension: \(1/6\) |
| 5 | \(0.1875\) | |
| 6 | \(0.2000\) | |
| 100 | \(0.2475\) | approaches \(1/4\) as \(D\to\infty\) |
Regard \(\Omega\) as a field and the Einstein action is the action of a conformally coupled scalar multiplied by an overall minus sign. That is why the kinetic term has the wrong sign — this is the conformal factor problem (Gibbons–Hawking–Perry 1978).
03The core — just make the wrinkles finer and it diverges
the weight \(e^{-S}=e^{+|S|}\) measured in bits is \(|S|/\ln2\) (with \(L=10\,\ell_P\), \(a=0.1\))
| \(n\) (number of wrinkles) | \(|S|\) | weight [bits] |
|---|---|---|
| 1 | \(2.36\) | \(3.4\) |
| 2 | \(9.42\) | \(13.6\) |
| 5 | \(58.9\) | \(85.0\) |
| 10 | \(235.6\) | \(339.9\) |
| 20 | \(942.5\) | \(1359.7\) |
| 50 | \(5890.5\) | \(8498.2\) |
The main point of this episode
It grows as \(n^2\) and never stops.
Making the wrinkles finer makes the path-integral weight arbitrarily large — \(e^{-S}\) diverges.
On Episode 19's scale, there are directions in which the "surprise" is infinite.
Figure: the Euclidean action, and the path-integral weight, when the conformal factor is wrinkled. An ordinary scalar field rises, so its weight shrinks; the conformal factor falls the other way and its weight grows. Move the slider — there is no bottom anywhere.
04How many diverging directions are there?
The conformal factor carries one degree of freedom per point, so the action is unbounded below in every one of those \(5.2\times10^{182}\) directions. Logarithmically, that is 607 bits' worth of directions.
Conclusion of §04
It is the same currency as the \(3\times10^{122}\) bits of information Episode 24 counted the universe as having processed —
except that this counts not information, but broken directions.
05The fix — rotate the contour
This is the same shape of ledger as Episode 32's cosmon or Episode 29's MOND — a rule added by hand always carries a price.
06Where the ghost sits — the trade with Episode 34
| Theory | Spin 2 | Conformal factor | Prescription |
|---|---|---|---|
| Einstein gravity | healthy (right sign) | a ghost | rotate the contour |
| Conformal gravity (Ep. 34) | a ghost (fourth-order) | removed by gauge | unresolved |
Conclusion of §06
It was a trade: put the ghost in the conformal factor, or put it in the spin 2.
An option that puts it in neither has not been found.
── This is Part V's thesis itself: the place where the tool breaks does not vanish; it moves.
07\(D=2\) is the exception — and it connects to Episode 37
Episode 38 §07: quantum theory wrote a kinetic term into the "zero coefficient" (Liouville in two dimensions).
── Something that vanished classically growing under quantisation is not a one-off coincidence.
(1) §03's divergence is not a classical instability. In classical general relativity the conformal mode is fixed by the Hamiltonian constraint and is not a propagating degree of freedom (in four dimensions only the two spin-2 components propagate). What is broken is the definition of the Euclidean path integral, not the stability of stars or of spacetime.
(2) §03's numbers are a toy model: a single mode on a flat torus. The values \(L=10\,\ell_P\) and \(a=0.1\) carry no physical meaning — they are an example chosen to display the behaviour "grows as \(n^2\), without bound". Do not take the absolute bit counts seriously.
(3) §04's "\(5.2\times10^{182}\) directions" is a mode count with a Planck-length cutoff. It depends entirely on how the cutoff is drawn, and with the cutoff removed the number is infinite — it looks like a large finite number only because a cutoff was imposed, and it indicates scale, not the severity of the problem.
(4) §01's sign depends on the metric signature convention and the overall sign of the action, and references write it as \(\pm6\) accordingly. What is convention-independent is the relation "the conformal factor's kinetic term has the opposite sign to a matter scalar's" and the magnitude \((D-1)(D-2)\).
(5) §05's contour rotation remains an active point of debate. Which contour to take in general is unsettled (prescriptions using Picard–Lefschetz theory have been discussed in recent years), and this document claims nothing beyond "it works at one loop".
(6) §06's "trade" is this series' reading. It has not been proved that no theory can avoid a ghost in both places — "none has been found within the known frameworks" is the accurate statement.
Exercises
- What is the coefficient of the conformal factor's kinetic term in \(D\) dimensions, and where does it vanish?
Show the answer
\((D-1)(D-2)\). It vanishes only at \(D=1\) and \(D=2\), and at \(D=4\) it is 6 — in our dimension there is no way out. - What does "the conformal factor is a conformally coupled scalar with the wrong sign" mean?
Show the answer
Regarded as a field, \(\Omega\) makes the Einstein action take the form of a conformally coupled scalar's action (\(\xi=(D-2)/4(D-1)\), which is \(1/6\) in four dimensions) multiplied by an overall minus. That flips the sign of the kinetic term, and the energy loses its lower bound. - Double the number of wrinkles \(n\). By what factor does the path-integral weight in bits grow?
Show the answer
By four (it goes as \(n^2\)). As the table shows, \(n=10\) gives 339.9 bits and \(n=20\) gives 1359.7 — with no upper bound. But per caveat (2), these are toy-model numbers. - Does this divergence mean astronomical objects are unstable?
Show the answer
No. In classical general relativity the conformal mode is fixed by the Hamiltonian constraint and does not propagate. What is broken is the definition of the Euclidean path integral, not the stability of spacetime. - (Harder) Putting Episode 34 and this episode together, what can be said about ghosts?
Show the answer
That they move rather than disappear. Einstein gravity has a healthy spin 2 and a ghostly conformal factor; conformal gravity removes the conformal factor by gauge but pays with a spin-2 ghost. An option that puts a ghost in neither has not been found within the known frameworks (caveat 6).
Summary: the ghost does not vanish, it moves
Write \(g=\Omega^2\hat g\), rewrite the Einstein action, and a kinetic term for the conformal factor \(\Omega\) appears. Its coefficient is \((D-1)(D-2)\), with the sign opposite to an ordinary scalar. It vanishes only at \(D=1\) and \(D=2\); at \(D=4\) it is 6 — our dimension has no way out. Regarded as a field, \(\Omega\) makes the Einstein action a conformally coupled scalar (\(\xi=1/6\)) with an overall minus.
How bad is it? Wrinkle a flat torus with \(\Omega=a\sin(2\pi nx/L)\) and the Euclidean action is \(|S|=2.357a^2n^2L^2\), so the weight \(e^{+|S|}\) grows as \(n^2\) forever — 340 bits at \(n=10\), 8498 bits at \(n=50\), with no bound. Making the wrinkles finer makes the path integral diverge. And there are \(5.2\times10^{182}\) Planck volumes inside the Hubble radius, so every one of them is an unbounded direction.
The fix is to rotate the contour for the conformal factor alone, \(\Omega\to i\Omega\) (Gibbons–Hawking–Perry). It does work at one loop, but there is no derivation from first principles — the same ledger as Episode 32's cosmon or Episode 29's MOND, where a rule added by hand always carries a price.
And the most important thing this time. It was a trade: put the ghost in the conformal factor, or put it in the spin 2. Einstein gravity has a healthy spin 2 and a ghostly conformal factor; conformal gravity (Episode 34) removes the conformal factor by gauge and pays with a spin-2 ghost. An option putting it in neither has not been found — which is Part V's thesis itself: the place where the tool breaks does not vanish; it moves.
Finally, \(D=2\). The coefficient is zero, yet two-dimensional gravity does have a Liouville action — it grows out of the anomaly (Episode 37). Last time quantum theory wrote a number into the "zero column"; this time it writes a kinetic term into the "zero coefficient" — something vanishing classically and growing under quantisation is not a one-off coincidence.
This document is Episode 38 of "c·t = const, That Clicks" (the second of Part V), written for physics-minded high-school and university readers. The conformal factor problem has been a well-known standard issue since Gibbons, Hawking & Perry (1978, Nucl. Phys. B138, 141) and nothing here is a new claim — the numbers are computed in kenshou/calc42.py. §03's divergence is not a classical instability — in classical general relativity the conformal mode is fixed by the Hamiltonian constraint and does not propagate (only the two spin-2 components do in four dimensions), so what is broken is the definition of the Euclidean path integral, not the stability of spacetime. §03's numbers are a toy model of a single mode on a flat torus, and \(L=10\,\ell_P\), \(a=0.1\) carry no physical meaning — they display the behaviour "grows as \(n^2\), without bound", so the absolute bit counts should not be taken seriously. §04's \(5.2\times10^{182}\) is a mode count with a Planck-length cutoff and depends entirely on how the cutoff is drawn; with the cutoff removed the number is infinite — it indicates scale only. §01's sign depends on the metric signature and the overall sign of the action and is written as \(\pm6\) in different references — what is convention-independent is the relation "opposite in sign to a matter scalar" and the magnitude \((D-1)(D-2)\). §05's contour rotation remains an active point of debate; which contour to take in general is unsettled (prescriptions using Picard–Lefschetz theory have been discussed in recent years), and nothing beyond "it works at one loop" is claimed here. §06's "trade" is this series' reading — it has not been proved that no theory can avoid a ghost in both places; "none has been found within the known frameworks" is the accurate statement. ── To make a PDF, use your browser's Print dialogue (sliders freeze and answers are hidden in the print version).