Thinking in Waves Episode 45 / Part XV — Making the Medium Ourselves
Looking at "nothing" as a wave
Shaking the Vacuum —
Heat turns out to be time measured logarithmically
The sum of zero-point energies diverges, but the divergent part is proportional to volume, so it subtracts away — what remains is \(-1/12\) (numerically \(-0.0833333\)).
★ Shake a boundary and photons come out of nothing (\(n=|\beta|^2\)). And \(|\alpha|^2-|\beta|^2=\det M=1\) — Episode 36, exactly.
★★ Why an accelerating observer sees the vacuum as hot: gamma function → \(\sinh\) → \(1/(e^x-1)\). Measure time logarithmically and a Planck distribution falls out.
Episode 44 got as far as time modulation making a wave grow.
★ So what happens when there is no wave at all to grow? That is this episode.
01Subtracting an infinity — possible because the divergence is a volume term
Between walls a distance \(a\) apart the modes are \(\omega_n=n\pi c/a\), and \(E=\frac\hbar2\sum\omega_n\) diverges. Insert a cutoff \(e^{-\delta n}\) and see what diverges.
| \(\delta\) | \(\sum n e^{-\delta n}\) (numerical) | \(1/\delta^2\) | ★ difference | \(-1/12\) |
|---|---|---|---|---|
| 0.1 | 99.91670832 | 100.00000000 | −0.08329168 | −0.08333333 |
| 0.01 | 9999.91666708 | 10000.00000000 | −0.08333292 | −0.08333333 |
| 0.003 | 111111.02777781 | 111111.11111111 | ★ −0.08333330 | −0.08333333 |
The point of §01
\(\sum n e^{-\delta n}=\dfrac1{4\sinh^2(\delta/2)}=\dfrac1{\delta^2}-\dfrac1{12}+O(\delta^2)\).
★★ The divergent term \(1/\delta^2\) is proportional to \(a\) — a volume term, identical with or without the walls. That is why it can be subtracted.
★ What remains is \(-1/12\) — finite, and negative.
★★ We did not "subtract an infinity"; we took the difference between having walls and not having them. That is the correct reading. As in Episode 7's "only dimensionless quantities are physics," only what can be compared carries meaning.
02The 3D electromagnetic case — how big is it really
| Separation \(a\) | \(|P|\) [Pa] | relative to 1 atm | |
|---|---|---|---|
| 1 mm | 1.3001e−15 | 1.283e−20 | |
| 1 μm | 1.3001e−3 | 1.283e−8 | |
| 100 nm | 1.3001e+1 | 1.283e−4 | |
| 10 nm | 1.3001e+5 | 1.283 | ★ exceeds atmospheric pressure |
The point of §02
\(P=-\pi^2\hbar c/(240a^4)\). ★ Close the gap to 10 nm and it is 1.28 atmospheres — the push of "nothing."
★ Being \(a^{-4}\), one decade closer is 10000 times stronger. Hence unmeasurable at 1 mm and a design consideration for MEMS at 100 nm.
03★ Shake it, and photons come out of nothing
From the one-period monodromy \(M\) we extract the Bogoliubov coefficients — \(\alpha=\frac{M_{11}+M_{22}}2-i\frac{M_{12}-M_{21}}2\), \(\beta=\frac{M_{11}-M_{22}}2+i\frac{M_{12}+M_{21}}2\).
| \(h\) | \(\det M\) | \(|\alpha|^2\) | \(|\beta|^2=n\) | \(|\alpha|^2-|\beta|^2\) |
|---|---|---|---|---|
| 0 | 1.00000000 | 1.00000000 | ★ 0.00000000 | 1.00000000 |
| 0.05 | 1.00000000 | 1.00154271 | 0.00154271 | 1.00000000 |
| 0.10 | 1.00000000 | 1.00617788 | 0.00617788 | 1.00000000 |
| 0.40 | 1.00000000 | 1.10115948 | ★ 0.10115948 | 1.00000000 |
Stack periods and the created photons grow exponentially (\(h=0.1\)).
| periods | \(n=|\beta|^2\) | \(|\alpha|^2-|\beta|^2\) |
|---|---|---|
| 1 | 0.006178 | 1.000000000 |
| 10 | 0.754098 | 1.000000000 |
| 20 | 5.290934 | 1.000000000 |
| 60 | 3089.030869 | ★ 1.000000000 |
The point of §03 — ★★★
\(|\alpha|^2-|\beta|^2=\det M=1\).
3089 photons later, that 1 has not moved.
★ Episode 36 said "\(\det M=1\) is the preservation of uncertainty"; this is its particle-number version.
★★ At \(h=0\), \(n=0\) — shake nothing and nothing comes out. What comes out is the shaker's work.
★ This section simply recounts Episode 44 §06's "inside the time gap it grows" in units of photons.
04Unruh — accelerate, and the vacuum looks warm
| acceleration \(a\) [m/s²] | \(T\) [K] | |
|---|---|---|
| 9.8 | 3.9739e−20 | surface gravity on Earth |
| \(10^{10}\) | 4.0550e−11 | an electron in a strong field |
| 2.47e+20 | 1.0016 | ★ \(T=1\) K |
| \(10^{26}\) | 4.0550e+5 | ★ 1350× room temperature |
The point of §04
★ Reaching 1 K takes \(2.5\times10^{20}\) m/s² — which is why it has never been measured directly.
★ Read the other way, this is the quantitative reason the vacuum is cold in everyday life — the same structure as Episode 5's "a period longer than the age of the universe is indistinguishable from a constant." Not absent; out of reach.
05★★ Why it becomes "heat" — logarithmic time and the exponential map
Fourier transform that in \(\eta\) and a gamma function appears.
| \(\Omega/a\) | \(|\beta/\alpha|^2=e^{-2\pi\Omega/a}\) | ★ \(n=1/(e^{2\pi\Omega/a}-1)\) |
|---|---|---|
| 0.1 | 5.334881e−1 | 1.143568 |
| 0.5 | 4.321392e−2 | 4.516571e−2 |
| 1.0 | 1.867443e−3 | 1.870937e−3 |
| 2.0 | 3.487342e−6 | 3.487355e−6 |
We check the gamma identity numerically too — \(|\Gamma(iy)|^2=\pi/(y\sinh\pi y)\).
| \(y\) | \(|\Gamma(iy)|^2\) (numerical) | \(\pi/(y\sinh\pi y)\) | ratio |
|---|---|---|---|
| 0.1 | 9.83738115e+1 | 9.83738115e+1 | 1.000000 |
| 1.0 | 2.72029055e−1 | 2.72029055e−1 | 1.000000 |
| 2.0 | 5.86676483e−3 | 5.86676483e−3 | 1.000000 |
The point of §05 — ★★★ the core of this episode
Time goes logarithmic ⟹ an exponential enters the phase ⟹ a gamma function ⟹ \(\sinh\) ⟹ \(1/(e^x-1)\).
"Heat" turns out to be a way of measuring time.
★ Episode 20 found that the \(c\cdot t=\)const coordinate system is logarithmic time. The \(at=e^{a\eta}\) here is exactly the same form.
★★ Episode 28 sorted "two sources of exponentials," one of them being addition turning into multiplication. Acceleration is the operation that turns adding time into multiplying it.
★ The temperature \(T=a/2\pi\) is literally the exponent of \(e^{-2\pi\Omega/a}\) — temperature, here, is the fineness of the logarithmic time scale.
06Temperature depends on who is looking
| Observer | What they see | In this series' words |
|---|---|---|
| at rest | vacuum (\(n=0\)) | Ep. 9: zero-point only |
| ★ uniformly accelerating | a thermal bath (\(T=a/2\pi\)) | ★ looking in logarithmic time |
| ★ shaking a mirror | photons appear | ★ parametric, Episodes 40 and 44 |
The point of §06
★★ "How many particles are there" turned out to be a coordinate-dependent quantity.
★ Seen as waves this is unsurprising — it is a question of which basis you expand in. The "number of components" changes between a Fourier basis and a wavelet basis (Episode 22) in exactly the same way.
★★ The caution of Episodes 7 and 15 applies again — never mistake a coordinate-dependent quantity for physics.
07Still, nothing comes for free
| Gatekeeper | How it bites here |
|---|---|
| Passivity (Ep. 17, 18) | the work of shaking the mirror is the photons' energy |
| \(\det M=1\) (Ep. 36) | ★ \(|\alpha|^2-|\beta|^2=1\) — what grows comes in pairs |
| Sum rule (Ep. 44) | gain at one frequency is loss at another |
The point of §07
★ The Casimir force is a gradient of a function of position, not a stock of extractable energy. Once the plates have snapped together, separating them costs exactly the same work.
★★ The photons of the dynamical Casimir effect are likewise just the shaker's work coming back out.
★ Whenever you hear "extracting vacuum energy," look first for who is doing work — one of these three gatekeepers is always in the way.
08What wave language can and cannot say
| Item | Verdict | Basis |
|---|---|---|
| The divergence is a volume term | ◎ numerics | subtract \(1/\delta^2\), get \(-1/12\) |
| Casimir pressure \(\propto a^{-4}\) | ◎ numerics | 1.28 atm at 10 nm |
| ★ Shaking gives \(n=|\beta|^2\) | ◎ numerics | ★ \(n=0\) at \(h=0\) |
| ★ \(|\alpha|^2-|\beta|^2=1\) | ◎ numerics | ★ Episode 36 in photon number |
| Unruh temperature | ◎ numerics | \(2.5\times10^{20}\) m/s² for 1 K |
| ★ Heat is logarithmic time | ◎ analytic | ★ \(\Gamma\to\sinh\to1/(e^x-1)\) |
| The \(|\Gamma(iy)|^2\) identity | ◎ numerics | ratio 1.000000 |
| ★ No free energy | ◎ logic | passivity, \(\det M=1\), sum rule |
| ★ The cosmological constant problem | × no | ★ nothing to subtract against |
The point of §08 — ★ the caveat that matters most
§01 said the divergence can be subtracted, but only because we took the difference between having walls and not.
★★ Gravity responds to the absolute value, not a difference — so asking for the vacuum's energy density itself leaves nothing to subtract against.
★ That is the cosmological constant problem, and these tools cannot touch it at all. ★ The argument "the Casimir effect was measured, therefore vacuum energy is real" stops right here — what was measured is a difference.
Exercises
- In §03, 3089 photons appeared and \(|\alpha|^2-|\beta|^2=1\) never moved. What is being protected?
Show answer
The commutation relation — that is, uncertainty. \(|\alpha|^2-|\beta|^2=1\) is the condition for a Bogoliubov transformation to be canonical, and classically it is \(\det M=1\) (phase-space area).
★ It is the same equation as Episode 36's "squeeze all you like, \(\Delta X_1\Delta X_2=1/4\) does not move."
★★ So dynamical-Casimir photons necessarily come in pairs (\(\alpha\) grows whenever \(\beta\) does) — you cannot grow one alone. ★ The correlated photon pairs seen in experiment are a direct consequence of this identity. - §05 said heat is logarithmic time. Does anything look hot if you measure time logarithmically?
Show answer
No — logarithmic time alone is not enough. What is also needed is that the coordinate system sees only part of the vacuum. An accelerating observer has a horizon and cannot see beyond it.
★ Discard (trace out) the unseen part and what remains is a mixed state — that is why it looks thermal. Logarithmic time appears as the form of the coordinate change that creates the horizon.
★★ So using logarithmic time in Episode 20's \(c\cdot t=\)const system does not by itself produce heat. ★ What is required is an unseen region — entropy appears in proportion to the information thrown away (Episode 21). - §02's Casimir pressure went as \(a^{-4}\). Where does that exponent come from?
Show answer
Pure dimensional analysis. The only ingredients are \(\hbar,c,a\). Pressure has dimensions [energy]/[length]³ and \(\hbar c\) is [energy]×[length], so \(P\propto\hbar c/a^4\) — everything but the coefficient is one line.
★ On Episode 43's single table of orders, \(a^{-4}\) is a consequence of three dimensions. In \(d\) dimensions \(P\propto\hbar c/a^{d+1}\) — and §01's one-dimensional case gave \(F\propto1/a^2\), consistent.
★★ Only the coefficient (\(\pi^2/240\)) comes from "how the waves are counted" — the exponent is set by dimension, the coefficient by mode counting. Exactly the division of labour of Episode 30.
Summary — heat is time measured logarithmically
The sum of zero-point energies diverges, but the divergent term \(1/\delta^2\) is proportional to volume, so taking the difference between having walls and not removes it — leaving \(-1/12\) (numerically \(-0.0833333\)). The right reading is not "we subtracted an infinity" but "we took a difference." In one dimension, \(E=-\pi\hbar c/(24a)\), and the force is attractive.
For the 3D electromagnetic field, \(P=-\pi^2\hbar c/(240a^4)\) — 1.28 atmospheres at 10 nm. Being \(a^{-4}\), one decade closer is 10000 times more.
Shake a boundary and photons come out of nothing. \(n=0\) at \(h=0\); \(n=0.101\) at \(h=0.4\); \(n=3089\) after 60 periods.
★★★ And 3089 photons later, \(|\alpha|^2-|\beta|^2=\det M=1\) has not moved —
the particle-number version of Episode 36's "\(\det M=1\) preserves uncertainty."
Accelerate and the vacuum looks warm (\(T=\hbar a/2\pi ck_B\)). 1 K needs \(2.5\times10^{20}\) m/s² — out of everyday reach.
★★★ And that "heat" is —
the accelerating observer's time going logarithmic (\(at=e^{a\eta}\)) ⟹ an exponential in the phase
⟹ a gamma function ⟹ \(\sinh\) ⟹ \(1/(e^x-1)\).
The temperature \(T=a/2\pi\) is literally the exponent of \(e^{-2\pi\Omega/a}\).
We checked it numerically too — \(|\Gamma(iy)|^2=\pi/(y\sinh\pi y)\), ratio 1.000000. ★ Episode 20's "logarithmic time" and Episode 28's "addition becomes multiplication" meet here.
★ And nothing comes for free — passivity, \(\det M=1\), and the sum rule all stand in the way. The Casimir force is a gradient, not a stock; dynamical-Casimir photons are the shaker's work.
★★ The caveat that matters most: §01 worked because we took a difference, and gravity responds to the absolute value. So the argument "Casimir was measured, therefore vacuum energy is real" stops here, and the cosmological constant problem is entirely beyond these tools.
★ Next time those two meet head-on — black hole evaporation, and where the information went.
★★ The Hawking temperature is the same formula as §04's Unruh temperature with the surface gravity in place of \(a\). And Episode 39's holography returns here as "information is counted by area."
This document is Episode 45 of the "Thinking in Waves" series. Computations are in kenshou/shinku.js. On the numbers here: the cutoff mode sum and extraction of \(-1/12\), the Casimir pressures, the Bogoliubov coefficients and photon numbers from the monodromy matrix, the Unruh temperatures, and the \(|\Gamma(iy)|^2\) identity were all computed here. The Casimir effect and its \(a^{-4}\) law, zeta regularization (\(\zeta(-1)=-1/12\)), Bogoliubov transformations and \(|\alpha|^2-|\beta|^2=1\), the dynamical Casimir effect, the Unruh effect and \(T=\hbar a/2\pi ck_B\), and the Rindler computation in which a gamma function yields the Planck distribution are all well-known results and none are discoveries of this article — what this article did was re-shelve them under two of this series' own tools, "logarithmic time" (Episode 20) and "\(\det M=1\)" (Episode 36). §03 is a single-mode minimal model; the real dynamical Casimir effect is a multimode problem (it has been reported in superconducting-circuit experiments, which this article does not reproduce). The \(-1/12\) of §01 is the value from one particular regularization (an exponential cutoff); that the answer is regularization-independent is established in the literature but not checked here. As stated in §08, this article could only handle differences of energy, never the vacuum energy density itself (the cosmological constant problem). — To print, use your browser's Print and "Save as PDF."