In Episodes 4, 5, and 9 we saw “erasable forces” → Episode 12: so, how far can we erase, and what remains?
Over eleven episodes, we’ve peeled back the true nature of force one layer at a time. Inertial forces are the balancing of coordinate accounts (Episode 4), gravity is geometry (Episode 5), a real force is a give-and-take between fields (Episode 6), and at the root, it’s the connection of a local symmetry (Episode 9). The more we peeled, the stronger the hint grew: “force — couldn’t it be erased?” This next-to-last episode surveys the whole from the summit of the journey — how far can force be erased? And what is it that absolutely will not vanish? The answer folds the entire series into a single line — what can be erased is bookkeeping (gauge, coordinates); what can’t is curvature (the body of the relationship itself).
First, let’s recall the forces we really did erase. The inertial force (Episode 4) was the balancing of accounts for choosing an accelerating coordinate system, so it vanished once you returned to an inertial frame. Gravity (Episode 5) vanished locally into weightlessness once you chose a freely falling coordinate system. Both vanish by “re-attaching coordinates (the point of view).” In the language of Episode 9, these are components that vanish when you swap the gauge (the local choice).
Inertial force: vanishes when you return to an inertial frame.
Gravity (in part): at that one point, it vanishes when you move to a freely falling frame.
The common thread — an “apparent” component that appears or vanishes depending on the choice of coordinates or gauge. It is not the body of the physics.
Another form of “erasing” is unification. In Episode 8 we saw the three couplings of the electromagnetic, weak, and strong forces approach a single point at high energy. If they really do meet at one point, the three are branches of a single force, and the very distinction of “three” is a low-energy appearance — at high energy it vanishes. As we saw in Episode 9, all four forces were the same mold, “the connection of a local symmetry.” Even the “multiplicity of kinds” of force can be reduced (erased) under a deeper symmetry. Four, into a single principle.
So can everything be erased? No. Recall the tidal force foreshadowed in Episode 5. Erase gravity with free fall, and yet over a wide region the direction of gravity differs from place to place, so an object is stretched and squeezed. This tidal effect = the curvature of spacetime cannot be erased by any choice of coordinates. The same thing exists in electromagnetism — you can make the field zero at one point, but the “wrap-around” when you go once around a loop (the field strength = the curvature) is invariant even if you change the gauge.
Force is made of two layers.
① The locally erasable component (appears and vanishes with the swap of coordinates or gauge = bookkeeping).
② The uneraseable curvature (the wrap-around of a loop, the tidal effect. It remains under any point of view = the body of the physics).
The answer to “can we erase force entirely?”: bookkeeping can be erased; curvature cannot.
This is the point the main run reaches. In Episode 1 we said “force is the name of a relationship,” and in Episode 9 we made it concrete as “the connection demanded by local freedom.” Turn one more page at the end, and — that connection (force) split into a “bookkeeping” part that vanishes if you change your point of view and a “curvature” part that won’t vanish no matter what. Much of what we call and feel as “force” is actually the former (the balancing of coordinate and gauge accounts), and can be erased. The uneraseable latter (curvature) is precisely the physical body of force — the core of the relationship that truly exists between the two parties.
Unification is influential but not established. The electromagnetic and weak forces have been demonstrated as the electroweak unification, but grand unification including the strong force (Episode 8) has no decisive evidence such as proton decay and is unconfirmed. And no one has yet completed putting gravity into the same frame as the other three (quantum gravity). Gravity resembles the others in that it can be erased locally (the equivalence principle), but when made into a quantum theory it is not renormalizable — the “door left open” of the finale of the sister series “Cosmology That Clicks.”
The “erasable / uneraseable” dichotomy is a plain way of stating the relationship between the gauge field (the connection) and the curvature (the field strength); mathematically it corresponds to the fact that the gauge can be brought locally to a standard form and that the curvature is a gauge invariant. The figure is a conceptual model simplifying the relationship between a gravitational field and the tidal effect.
How far can force be erased? Inertial forces vanish locally with coordinates and gravity with free fall (Episodes 4&5), and the four forces can be reduced in kind through unification (Episodes 8&9). All of these are “bookkeeping components that appear and vanish with the swap of coordinates or gauge.” But the tidal effect that won’t vanish even in free fall = curvature, and the wrap-around of a loop that remains even after you zero it at one point = the field strength, won’t vanish under any point of view.
Force has two layers — erasable bookkeeping (gauge, coordinates) and uneraseable curvature (the body of the physics). Episode 1’s “force is the name of a relationship” is here resolved into “a relationship that splits into a part that vanishes with the point of view and a part that remains.” Much of the force we feel is bookkeeping, and can be erased. The uneraseable curvature is precisely the core of the relationship. — Though full unification and quantum gravity are still a dream in progress. The finale wraps up the meaning of this journey.
Print / save as PDF: ⌘+P (Ctrl+P on Windows). On screen, the slider lets you see how the central force can be erased while the tidal effect (curvature) remains. “See the answer” opens each solution.