Speed ①: a change travels at c → Speed ②: why "exactly" c
In Speed ① we saw that "a change in force travels at the speed of light \(c\)." But why exactly \(c\)? And what happens when the carrier is heavy, as with the weak force? Here we clearly dissect the most common mix-up ── "range (how far it reaches)" and "speed (how fast it travels)" are different things. The carrier's mass sets the range, but the speed of a change is kept at \(c\) as a ceiling for any carrier. Meeting Yukawa again from Episode 6 (\(\lambda=\hbar/mc\)), we separate these two.
When the carrier of a force (Episode 6) has zero mass, the change in its field travels at exactly the speed of light, and the force reaches out to infinity. The photon (electromagnetism), the gluon, and the graviton (hypothetical) are these. In Episode 6's \(\lambda=\hbar/mc\), if \(m\to0\) then \(\lambda\to\infty\) (infinite range), and Episode 10's inverse-square \(1/r^2\) was also a consequence of this "zero mass + long range." So both electromagnetism and gravity reach far, and their change chases you at \(c\) (the Sun story from Speed ①). Zero mass ↔ speed \(c\) ↔ infinite range is one packaging.
So what happens when the carrier is heavy, as with the weak force (W, Z)? As Episode 6 says, the range \(\lambda=\hbar/mc\) becomes much shorter (W and Z are about 90 times heavier than a proton, so the force reaches a range far smaller than an atomic nucleus). Here is where many people go wrong ── "short range = travels slowly." This is a mistake. Even with a heavy carrier, the speed at which the leading edge of the change (the news) advances still does not exceed \(c\); effectively it is \(c\). What mass changes is how far it reaches (range), not how fast it travels (speed).
What the carrier's mass \(m\) sets is the range \(\lambda=\hbar/mc\) (how far it reaches).
The speed at which a change travels is capped at \(c\) regardless of mass (causality, Speed ①).
So "the weak force is short-range," but that doesn't mean "the weak force is slow." Near does not mean slow.
In the figure below, try changing the carrier's mass. Raise the mass and the force's envelope (the range it reaches) shrinks visibly, yet the change's leading edge (the position of the news) stays in the same place ── that is, the speed doesn't change. You can see range and speed move independently.
Intuitively ── range is about "how much force a static source can keep exerting on its surroundings." A heavy carrier can "borrow" time only briefly (the uncertainty of Episode 6) and cannot travel far on its errand, so the range is short. Speed, on the other hand, is about "when a change occurs at the source, how fast that news gets there." The news (information), whether the carrier is heavy or light, cannot cross the causal wall of \(c\). How far (range) and how fast (speed) are, from the start, different questions.
The conclusion of The Speed of Force ②. A change in force is exactly \(c\) because the carrier has zero mass. When the carrier is heavy, the range shrinks, but the speed of the change stays \(c\). "Zero mass ↔ \(c\) ↔ infinite range" is one package, and giving it mass shrinks only the range, leaving the speed untouched. The mix-up that "the weak force is short-range, so it's slow" confuses range and speed ── these two were different questions from the start.
Strictly, the real particles of a heavy carrier (actual W and Z particles) have mass, so they move slower than light. However, the speed at which the "leading edge (the front of the signal)" of a change in force advances (the front velocity) does not exceed \(c\) even with mass; \(c\) is the ceiling. "A change is at most \(c\) (it does not cross the wall of effectively \(c\))" is the accurate way to say it, and the conclusion that "short range ≠ slow" is unchanged (we don't go into the distinctions among wave speeds like phase velocity and group velocity here).
The figure is a schematic reproducing only the key point that "the range shrinks with mass, but the position of the change's leading edge (= the speed) is unchanged"; it is not a numerical computation of actual field propagation. The leading edge's position is drawn as the point reached after a fixed time.
A change in force travels at exactly \(c\) because the carrier has zero mass (photon, gluon, graviton) ── zero mass ↔ speed \(c\) ↔ infinite range is one package. When the carrier is heavy (W, Z), the range \(\lambda=\hbar/mc\) becomes short, but the speed of a change is kept at \(c\) as a ceiling. Mass sets the range, \(c\) sets the speed ── different questions. "Short-range, so slow" was a confusion of range and speed.
Episode 6 (Yukawa's range), Episode 10 (inverse-square), and Speed ① (a change is c) meshed here into one. Next time, the subtlest and most fascinating story ── "a static force carries no news." The force of a source moving at constant velocity appears to "track its current position instantly." But it carries no information and breaks no causality. Resolving virtual particles and the true nature of the apparent instantaneity, we close the trilogy on speed.
Print / Save as PDF: ⌘+P (Ctrl+P on Windows). On screen, use the slider to watch how raising the mass shrinks the range while the leading edge (the speed) doesn't move. Click "See the answer" to open each solution.