Imagine striking a golf ball with a club, sending it flying. Being perfectly rigid, the golf ball would have to move off without any change of shape: all parts of the ball must start moving together. But now we run into a snag. No force can travel faster than light, so the blow delivered to one side of the ball can’t be felt by the other side until at least the time that light would take to traverse the
ball. Consequently the struck side would have to start moving before the remote side. But then the ball would have to change shape — it would be compressed. It follows that the ball must have at least a certain amount of squashiness: perfectly rigid bodies are inconsistent with the theory of relativity. But if an electron can be squashed, then it can also be stretched — and, if assaulted violently enough, pulled apart. So a little golf-ball electron couldn’t be a truly elementary body either.
But what if we imagine the little ball shrunk to a single point? Light would then take no time at all to traverse the (zero) distance across it. Unfortunately that solves one problem only to create another. There is electric charge distributed through the little ball. Imagine trying to shrink the ball, complete with its resident charge, to a smaller and smaller radius. To compress the charge into ever-smaller volumes requires the expenditure of energy to overcome the electrical repulsion. According to the inverse square law of electric force discovered in the eighteenth century by Charles Coulomb,4 the repulsion between the parts of the ball rises without limit as the charge is confined to ever-smaller volumes. An infinite amount of energy would be needed to compress the ball to zero radius, and this energy would be stored inside the electron. Taking into account Einstein’s formula E = mc^2 ,an infinite internal energy has the nonsensical implication that the electron should have an infinite mass. So we are left with a dilemma: the electron can be neither a point nor a finite ball without coming into flagrant contradiction with reality.
Now, you might think that quantum mechanics would come to the rescue here. By smearing out the spatial location of a pointlike particle, it would seem to circumvent the difficulty that all portions of the electric charge are accumulated at a single place. In fact, quantum mechanics makes the problem even worse. To get some idea of why, remember how electric forces are conveyed in quantum mechanics — by the exchange of photons (see Figure 17, p. 96). The same forces will also act between the various bits of charge distributed through the “little ball,” implying that a swarm of virtual photons surrounds and interpenetrates the electron. A calculation shows that the swarm’s energy gets bigger as the size of the electron gets smaller, because the close-in virtual photons are the most energetic. The total energy of the swarm rises to infinity as the radius of the electron is shrunk to zero. It doesn’t matter that the overall spatial location of the electron might be ill defined: wherever it is, the cloud is there with it, clothing it with limitless amounts of energy, and hence mass.
What are we to make of this? By using mathematical tricks, physicists are able to dodge round the infinities and still use the theory of quantum electrodynamics to obtain sensible answers to questions about particle masses, energy levels, scattering processes, and so on. The theory remains brilliantly successful. But the fact that infinities occur is a worrying symptom that something is deeply wrong, something that needs fixing.
The same general analysis can be applied to the gravitational field. Shrinking a ball to zero radius would involve infinite gravitational energy. Quantum mechanically, the gravitational force is conveyed by gravitons, and the gravitational field surrounding a particle can be envisaged as a cloud of virtual gravitons. As in the electromagnetic case, infinities follow. But with gravitation there is double trouble. Any pointlike particle (for example, an electron) would be surrounded by a virtual graviton cloud containing infinite energy.
But because energy is a source of gravitation, gravitons themselves contribute to the total gravitational field. (In effect, gravity gravitates.) So each virtual graviton in the cloud surrounding the central particle possesses its own cloud of yet more gravitons clustering around it, and so on ad infinitum: clouds around clouds around clouds ... and each cloud contains infinite energy! This time the infinities can’t be so easily dodged. A straightforward quantum description of the gravitational field produces a limitless progression of infinities, ruining any hope of making sensible predictions from the theory.
