Entering head-on the "edge of the band = Planck scale" that kept appearing in Bonus 4, Ep. 5, and Ep. 6
Throughout this series we've pointed to the same place again and again: Bonus 4 said "equivalence breaks at the Planck scale," Episode 5 said "the difference matters at the edge of the band," Episode 6 said "the biggest remaining door is gravity." The finale enters that place — the Planck scale — head-on. In Episode 6, the three forces (electromagnetic, weak, strong) could be run toward one point at high energy. So can the fourth — the most familiar yet most troublesome, gravity — fit into the same picture? To cut to it: no one has managed it yet. Witnessing why it doesn't fit, using the series' tools, is the last step of this journey.
The three forces' couplings in Episode 6 (\(\alpha\), etc.) shared an important trait — each was a plain, unit-free number. That's why, even as they ran, they could be compared as "numbers" and gathered to one point. But Newton's constant \(G\), which measures the strength of gravity, is not like that.
Electromagnetic/weak/strong couplings → unit-free numbers (e.g. \(\alpha\approx1/137\)).
Gravity's coupling \(G\) → carries units (in natural units, the dimension of "one over energy squared").
As Bonus 2 showed, a quantity with units can't say "strong" or "weak" on its own. To make the strength a dimensionless number, you must multiply \(G\) by "energy squared" to cancel the units. So gravity's effective strength is —
Here appears a property decisively different from the other three forces. Gravity's strength grows as the square of energy \(E\). At low energy (the everyday world), \(E^2\) is small, so gravity is absurdly weak compared to the other forces (this is why a magnet can beat the whole Earth's gravity to lift a nail). But raise the energy and \(E^2\) kicks in hard, and gravity's strength shoots up.
Put energy on the horizontal axis and "force strength" on the vertical. The other three forces (blue), as we saw in Episode 6, only drift gently as they run. But gravity (purple) shoots up as \(E^2\) and, at a certain point, catches up to the others. That catch-up point is —
The energy where gravity's strength becomes 1 (comparable to the other forces)
$$G\cdot E^2 \sim 1 \quad\Rightarrow\quad E \sim \frac{1}{\sqrt{G}} \equiv M_{\text{Planck}} \approx 10^{19}\ \text{GeV}$$This \(M_{\text{Planck}}\approx10^{19}\) GeV is the Planck scale — a thousand times higher still than Episode 6's grand unification (\(10^{16}\) GeV). Here gravity finally stands shoulder to shoulder with the other three, and all four forces reach the same strength — the highest-energy stage in the universe, where "unification of all four forces" is hoped for. This is the identity of the place we've called "the edge of the band" in Episode 5 and Bonus 4.
If gravity merely got stronger at high energy, that would be fine. The real problem lies beyond. In Episode 6, force theories could cleanly handle the infinities appearing in calculations by "running (renormalizing)." But try to do the same with gravity and —
Because \(G\) carries units (one over energy squared), every time you make the calculation finer, new kinds of infinity appear, one after another. And to cancel them, you must bring in an infinite number of new unknown quantities each time. Since you can't fix infinitely many numbers by experiment, the theory loses its power to predict. This is "gravity is non-renormalizable," the central difficulty of quantum gravity.
The other three forces: couplings are dimensionless → infinities can be absorbed by a finite number of quantities → predictions possible.
Gravity: the coupling \(G\) carries dimensions → infinities demand infinitely many new quantities → predictive power lost.
What's interesting is that this is the flip side of what Episodes 6 and 2 have said all along. What truly matters in physics was the dimensionless ratio (like \(\alpha\)). Gravity's difficulty, too, is rooted in "\(G\) not being dimensionless." Quantities that carry dimensions can't be handled straightforwardly — the series' backbone stands in the way, in the same form, even at the very far edge.
Here we return to the series' star, \(\alpha\). In Episode 6, \(\alpha\) was a running number. But that running was calculated while ignoring gravity. Approach the Planck scale and gravity, growing as \(E^2\), can no longer be ignored, and a contribution from gravity enters \(\alpha\)'s running too. And since gravity can't be renormalized, that contribution can't be properly calculated in current physics.
In other words — the final answer to the question reposed in Episode 6, Part 1, "where do \(\alpha\)'s running and the mechanism that fixes it come from?", is likely hidden inside the quantum gravity of the Planck scale. And no one has yet set foot there. When you first placed "\(\alpha\) as the first constant," that intuition, pushed to its limit, led right to the entrance of physics' greatest unsolved problem — "if we could build an ultimate theory including gravity, maybe we could also produce the value of \(\alpha\)."
What's certain in this episode goes only as far as "why gravity doesn't fit into the same frame as the other three (\(G\) carries dimensions and can't be renormalized)." Beyond that — what actually happens at the Planck scale, whether space becomes discrete there, whether \(\alpha\)'s value is fixed there — no one yet knows. Superstring theory and loop quantum gravity are candidates, but decisive experimental backing is zero. The Planck scale is a thousand trillion times above today's accelerators, with no means to touch it directly.
So the series ends honestly here — the last door of "Cosmology That Clicks" is not yet open. But that's not a failure; it's the living frontier that the enterprise of physics is, right now, trying to push open by human hands.
The electromagnetic, weak, and strong forces have dimensionless couplings and could be run toward one point (Episode 6). But gravity's coupling \(G\) carries dimensions, and its strength \(G\cdot E^2\) shoots up as the square of energy, catching the other forces at the Planck scale, \(10^{19}\) GeV — the identity of the "edge of the band" we'd pointed to in Episode 5 and Bonus 4. Yet because \(G\) carries dimensions, gravity can't be renormalized, demands infinitely many unknowns, and loses predictive power. So gravity still can't be fit into this picture.
And beyond the Planck scale hides the final answer to why \(\alpha\) has its value — your original intuition, "\(\alpha\) as the first constant," pushed to its limit, was the entrance to humanity's greatest unsolved problem. This series ends before an open door. Setting out from one guide-line — that light used to be faster — through the atom's 1/137, Schrödinger, and grand unification, we now stand before the farthest door: gravity. Beyond here, there's no map yet. Maybe you'll draw it.
Print / save as PDF: ⌘+P (Ctrl+P on Windows). On screen, the slider shows gravity catching up as E². "Show answer" opens the solutions.