The force of an electron on itself
The discrepancy between the two formulas for the electromagnetic mass is especially annoying, because we have carefully proved that the theory of electrodynamics is consistent with the principle of relativity. Yet the theory of relativity implies without question that the momentum must be the same as the energy times $v/c^2$. So we are in some kind of trouble; we must have made a mistake. We did not make an algebraic mistake in our calculations, but we have left something out. In deriving our equations for energy and momentum, we assumed the conservation laws. We assumed that all forces were taken into account and that any work done and any momentum carried by other “nonelectrical” machinery was included. Now if we have a sphere of charge, the electrical forces are all repulsive and an electron would tend to fly apart. Because the system has unbalanced forces, we can get all kinds of errors in the laws relating energy and momentum. To get a consistent picture, we must imagine that something holds the electron together. The charges must be held to the sphere by some kind of rubber bands—something that keeps the charges from flying off. It was first pointed out by Poincaré that the rubber bands—or whatever it is that holds the electron together—must be included in the energy and momentum calculations. For this reason the extra nonelectrical forces are also known by the more elegant name “the Poincaré stresses.” If the extra forces are included in the calculations, the masses obtained in two ways are changed (in a way that depends on the detailed assumptions). And the results are consistent with relativity; i.e., the mass that comes out from the momentum calculation is the same as the one that comes from the energy calculation. However, both of them contain two contributions: an electromagnetic mass and contribution from the Poincaré stresses. Only when the two are added together do we get a consistent theory.  It is therefore impossible to get all the mass to be electromagnetic in the way we hoped. It is not a legal theory if we have nothing but electrodynamics. Something else has to be added. Whatever you call them—“rubber bands,” or “Poincaré stresses,” or something else—there have to be other forces in nature to make a consistent theory of this kind. Clearly, as soon as we have to put forces on the inside of the electron, the beauty of the whole idea begins to disappear. Things get very complicated. You would want to ask: How strong are the stresses? How does the electron shake? Does it oscillate? What are all its internal properties? And so on. It might be possible that an electron does have some complicated internal properties. If we made a theory of the electron along these lines, it would predict odd properties, like modes of oscillation, which haven’t apparently been observed. We say “apparently” because we observe a lot of things in nature that still do not make sense. We may someday find out that one of the things we don’t understand today (for example, the muon) can, in fact, be explained as an oscillation of the Poincaré stresses. It doesn’t seem likely, but no one can say for sure. There are so many things about fundamental particles that we still don’t understand. Anyway, the complex structure implied by this theory is undesirable, and the attempt to explain all mass in terms of electromagnetism—at least in the way we have described—has led to a blind alley.  We would like to think a little more about why we say we have a mass when the momentum in the field is proportional to the velocity. Easy! The mass is the coefficient between momentum and velocity. But we can look at the mass in another way: a particle has mass if you have to exert a force in order to accelerate it. So it may help our understanding if we look a little more closely at where the forces come from. How do we know that there has to be a force? Because we have proved the law of the conservation of momentum for the fields. If we have a charged particle and push on it for awhile, there will be some momentum in the electromagnetic field. Momentum must have been poured into the field somehow. Therefore there must have been a force pushing on the electron in order to get it going—a force in addition to that required by its mechanical inertia, a force due to its electromagnetic interaction. And there must be a corresponding force back on the “pusher.” But where does that force come from? The picture is something like this. We can think of the electron as a charged sphere. When it is at rest, each piece of charge repels electrically each other piece, but the forces all balance in pairs, so that there is no net force. [See Fig. 28–3(a).] However, when the electron is being accelerated, the forces will no longer be in balance because of the fact that the electromagnetic influences take time to go from one piece to another. For instance, the force on the piece $\alpha$ in Fig. 28–3(b) from a piece $\beta$ on the opposite side depends on the position of $\beta$ at an earlier time, as shown. Both the magnitude and direction of the force depend on the motion of the charge. If the charge is accelerating, the forces on various parts of the electron might be as shown in Fig. 28–3(c). When all these forces are added up, they don’t cancel out. They would cancel for a uniform velocity, even though it looks at first glance as though the retardation would give an unbalanced force even for a uniform velocity. But it turns out that there is no net force unless the electron is being accelerated. With acceleration, if we look at the forces between the various parts of the electron, action and reaction are not exactly equal, and the electron exerts a force on itself that tries to hold back the acceleration. It holds itself back by its own bootstraps. It is possible, but difficult, to calculate this self-reaction force; however, we don’t want to go into such an elaborate calculation here. We will tell you what the result is for the special case of relatively uncomplicated motion in one dimension, say $x$. Then, the self-force can be written in a series. The first term in the series depends on the acceleration $\ddot{x}$, the next term is proportional to $\dddot{x}$, and so on.1 The result is \begin{equation} \label{Eq:II:28:9} F=\alpha\,\frac{e^2}{ac^2}\,\ddot{x}- \frac{2}{3}\,\frac{e^2}{c^3}\,\dddot{x}+ \gamma\,\frac{e^2a}{c^4}\,\ddddot{x}+\dotsb, \end{equation}  where $\alpha$ and $\gamma$ are numerical coefficients of the order of $1$. The coefficient $\alpha$ of the $\ddot{x}$ term depends on what charge distribution is assumed; if the charge is distributed uniformly on a sphere, then $\alpha=2/3$. So there is a term, proportional to the acceleration, which varies inversely as the radius $a$ of the electron and agrees exactly with the value we got in Eq. (28.4) for $m_{\text{elec}}$. If the charge distribution is chosen to be different, so that $\alpha$ is changed, the fraction $2/3$ in Eq. (28.4) would be changed in the same way. The term in $\dddot{x}$ is independent of the assumed radius $a$, and also of the assumed distribution of the charge; its coefficient is always $2/3$. The next term is proportional to the radius $a$, and its coefficient $\gamma$ depends on the charge distribution. You will notice that if we let the electron radius $a$ go to zero, the last term (and all higher terms) will go to zero; the second term remains constant, but the first term—the electromagnetic mass—goes to infinity. And we can see that the infinity arises because of the force of one part of the electron on another—because we have allowed what is perhaps a silly thing, the possibility of the “point” electron acting on itself.