The first half and the second half were different in kind ── and \(c\cdot t\) is what makes the seam visible
Bonus ① of the sister series "Temperature That Clicks" grew out of a reader's question ── "c shows up in both series; wouldn't a \(c\cdot t=\)constant view make things easier?" The answer there was "only where there is a horizon." Asked the same about this series, I counted again and found something bigger. There is a place where it works here too, and it works differently. The dividing line is not a horizon but ── whether what tunnels is a "particle" or "a shape extended through spacetime." Episodes 1–5 are particles. But the instantons, the Schwinger effect and the vacuum decay of Episodes 6–7 are objects with extension in Euclidean spacetime, whose action is a length, an area, a volume. And the light-speed expansion of the bubble ── which Episode 7 asked you to take on trust ── stops needing an explanation.
| Episode | What tunnels | Is \(c\cdot t\) useful? |
|---|---|---|
| 1, the wall | a particle in a potential (one variable) | no |
| 2, the Sun | a proton (non-relativistic Coulomb barrier) | no |
| 3, the crossover temperature | a particle crossing the inverted valley | no |
| 4, Josephson | the phase difference (one variable) | no |
| 5, superconductivity | (not tunneling) | no |
| 6, instantons | a field configuration extended in spacetime | needed |
| 6, the Schwinger effect | the worldline of a charge | very useful |
| 7, vacuum decay | a bubble extended in spacetime | most useful |
| Bonus ④, tunneling time | ── | useful, but as a constraint |
In Episodes 1–5, imaginary time \(\tau\) is a parameter for lining up paths. The action is \(S_E=\int d\tau\,[\frac12m\dot x^2+V]\), and no metric mixes \(\tau\) with \(x\). So multiplying by c only changes units.
In relativistic field theory, however, the Wick rotation turns the metric into \(+c^2d\tau^2+dx^2\). Now \(c\tau\) and \(x\) live in the same units, in the same space. And then the action can be written as a geometric quantity ── a length, an area, a volume. From there on it is a problem in figures, not in calculus.
Episode 6 merely listed \(e^{-\pi m^2c^3/eE\hbar}\) in a table. Let us derive it.
For a charge in a uniform electric field \(E\), the Euclideanised worldline action has two terms ── the length of the worldline and the area it encloses:
$$S_E=\underbrace{mc\oint ds}_{\text{length}}-\underbrace{\frac{eE}{c}\times(\text{enclosed area})}_{\text{work done by the field}}$$Try a circle of radius \(R\) in the \((x,\,c\tau)\) plane: \(S_E=2\pi R\,mc-\pi R^2\,eE/c\). Differentiate with respect to \(R\) and set to zero:
$$R=\frac{mc^2}{eE},\qquad S_E=\frac{\pi m^2c^3}{eE}\quad\checkmark$$The instanton is literally a circle, and the Schwinger exponent is that geometry. A line in Episode 6's table has become a figure.
The numbers close satisfyingly. Ask what field makes \(R\) equal the electron's reduced Compton wavelength \(\hbar/mc\), and it is exactly the critical field \(1.3\times10^{18}\) V/m. And there the exponent is exactly \(S_E/\hbar=\pi\). More generally,
$$\frac{S_E}{\hbar}=\pi\,\frac{R}{\hbar/mc}\qquad\text{── the exponent is }\pi\text{ times the radius in Compton wavelengths}$$This is the bigger one. In Episode 7 the bubble's budget was written in three dimensions:
$$\Delta E(R)=-\tfrac{4\pi}{3}R^3\varepsilon+4\pi R^2\sigma$$That is a correct intuition, and it gives the right critical radius \(R_c=3\sigma/\varepsilon\). But it is not the calculation of the action. What Coleman and Callan showed is that the minimum-action bounce is O(4)-symmetric in the four-dimensional Euclidean space \((c\tau,x,y,z)\) ── i.e. a four-dimensional sphere.
A 4-sphere of radius \(R\) has volume \(\pi^2R^4/2\) and surface (a 3-sphere) \(2\pi^2R^3\). So
$$S_E=\underbrace{2\pi^2R^3}_{\text{the 3-dimensional wall}}\sigma-\underbrace{\frac{\pi^2}{2}R^4}_{\text{the 4-dimensional interior}}\varepsilon$$Differentiating: \(6\pi^2R^2\sigma-2\pi^2R^3\varepsilon=0\) gives \(R=3\sigma/\varepsilon\) (the same as in three dimensions). Substituting back,
$$S_E=\frac{27\pi^2\sigma^4}{2\varepsilon^3}\quad\checkmark$$The expression Episode 7 simply quoted is derived here. And the strange coefficient \(27\pi^2/2\) turns out to come from the 4-sphere's volume \(\pi^2R^4/2\) and surface \(2\pi^2R^3\).
This is the best thing in the episode. Episode 7 said that a bubble past the critical radius expands at nearly the speed of light. That was a "please take this on trust."
Return the 4-sphere to real time ── i.e. undo the Wick rotation:
And in relativity, \(x^2-(ct)^2=\)constant is the worldline of an object with constant proper acceleration (see the sister series "Relativity That Clicks"). That is:
The bubble wall accelerates at a constant proper acceleration \(a=c^2/R\) and asymptotically approaches the speed of light.
This is not an extra assumption. It is simply what a slice through the 4-sphere looks like. "It expands at light speed" was a consequence of O(4) symmetry.
And more ── invert that wall's proper acceleration \(a=c^2/R\) and the distance to the wall's own horizon is \(c^2/a=R\). The bubble's radius is also the distance to its wall's horizon. Exactly the same structure as the Schwinger circle.
The figure is precisely that. Left is the Euclidean \((x,c\tau)\) plane, right is real time \((x,ct)\). Slicing the left circle at \(\tau=0\) gives the starting point of the right-hand hyperbola.
Move the time slider and the wall runs off to the right. Watch it approach the dashed 45-degree lines (the light cone) without ever reaching them. The speed is automatically below \(c\) and asymptotically equal to it. Not because it was built that way ── because that is what a hyperbola does.
There is a reason it is a sphere: the configuration that minimises the action is the symmetric one.
Intuitively ── for a given four-dimensional volume, the sphere has the smallest surface. A bounce gains by volume and loses by surface, so the shape that minimises surface for a given volume is selected. That is the 4-sphere, and in the thin-wall limit it has been proved to be the minimum (Coleman–Glaser–Martin).
And O(4) symmetry becomes O(3,1) symmetry when returned to real time ── the bubble expands in a Lorentz-invariant way. It looks like the same hyperbola from every inertial frame, which is why arguments about vacuum decay do not depend on the choice of coordinates.
Apply the three grades of "similar" from Episode 5 to the series itself.
| Episodes 1–5 | Episodes 6–7 | |
|---|---|---|
| what tunnels | a particle or a phase (one variable) | a shape extended in spacetime |
| imaginary time is | a parameter | a coordinate (it enters the metric) |
| the action is | \(\int d\tau[\frac12m\dot x^2+V]\) | a geometric quantity (length, area, volume) |
| where the \(2\pi\) and the coefficients come from | dynamics (harmonic motion) | geometry (circles, spheres) |
| relation to the horizon family (Unruh etc.) | grade 2 (different implementation of one framework) | grade 3 (genuinely the same family) |
The thirteen episodes bundled under the one word "tunneling" changed in kind between Episode 5 and Episode 6. The first half is non-relativistic dynamics; the second half is the geometry of Euclidean spacetime. Only with the \(c\cdot t\) view does the seam become visible.
The backbone (what looks impossible in real time is ordinary in imaginary time) held all the way through, but where imaginary time was promoted from "parameter" to "coordinate," the character of the story changed.
Established: the worldline-instanton derivation of the Schwinger effect (a circular solution, \(R=mc^2/eE\), \(S_E=\pi m^2c^3/eE\)); that the Coleman–Callan bounce is O(4)-symmetric in the thin-wall limit, giving \(R=3\sigma/\varepsilon\) and \(S_E=27\pi^2\sigma^4/2\varepsilon^3\) from the 4-sphere's volume and surface; that the Euclidean circle \(x^2+(c\tau)^2=R^2\) becomes the hyperbola \(x^2-(ct)^2=R^2\) in real time, which is the worldline of constant proper acceleration \(a=c^2/R\); that the bubble wall therefore approaches the speed of light asymptotically; that O(4) symmetry becomes O(3,1) in real time; and the O(4) minimality in the thin-wall limit (Coleman–Glaser–Martin). All standard results.
Points to note: (1) This bonus too merely restores derivations omitted in the main series; it is not new physics. What it buys is perspective, not content. (2) Episode 7's three-dimensional energy budget is not wrong ── the critical radius \(R=3\sigma/\varepsilon\) comes out the same in three or four dimensions. Only the value of the action and the coefficient \(27\pi^2/2\) require four dimensions. Episode 7's presentation was loose rather than incorrect. (3) O(4) is not always the minimum. It is proved in the thin-wall limit, but in general a solution with different symmetry can dominate. In particular at finite temperature an O(3)×S¹-symmetric bounce (i.e. thermal nucleation) takes over, which is the field-theory version of Episode 3's "climb over, or slip through." (4) Curved spacetime changes things. Including gravity can make the bubble interior an open universe and alter the nucleation rate (Coleman–De Luccia). The body treats flat spacetime. (5) That the Schwinger circle's radius equals the horizon distance is a geometric fact, but it is not a claim that the Schwinger and Unruh effects can be identified. There is literature relating them, and it is not a simple equality. (6) The Schwinger effect has not been observed (as Episode 6's table notes, the critical field \(1.3\times10^{18}\) V/m is out of reach).
In Bonus ① of "Temperature That Clicks," \(c\cdot t\) helped where there was a horizon. Here the dividing line is different ── whether what tunnels is a "particle" or "a shape extended through spacetime." Episodes 1–5 are particles, and c never appears. Episodes 6–7 were another matter.
The Schwinger effect was a circle. The Euclidean worldline action is "length minus enclosed area," and a circle of radius \(R=mc^2/eE\) gives \(S_E=\pi m^2c^3/eE\). And \(R=c^2/a\) ── exactly the horizon distance of the accelerating charge, linking to Bonus ① of Temperature.
Coleman's bounce was a four-dimensional sphere. From volume \(\pi^2R^4/2\) and surface \(2\pi^2R^3\) come \(R=3\sigma/\varepsilon\) and \(S_E=27\pi^2\sigma^4/2\varepsilon^3\) ── the expression Episode 7 merely quoted, now derived. And as a bonus the light-speed expansion stops needing an explanation: returning the sphere to real time gives the hyperbola \(x^2-(ct)^2=R^2\), the worldline of constant proper acceleration \(a=c^2/R\), so it approaches \(c\) asymptotically without exceeding it. Not an extra assumption but a consequence of O(4) symmetry.
And the classification can be redrawn ── the thirteen episodes changed in kind between Episode 5 and Episode 6. In the first half imaginary time is a parameter and coefficients come from dynamics (grade 2 relative to the horizon family); in the second half it is a coordinate and the action is geometric (grade 3). The backbone held all the way, but there was an invisible seam where imaginary time got promoted.
Print / make a PDF: ⌘+P (Ctrl+P on Windows). On screen the time slider sends the wall running, approaching the 45-degree light cone without ever reaching it. The radius slider gives the corresponding field, proper acceleration and horizon temperature. "See the answer" opens each solution.