The A B C of atoms

by Bertrand Russell

It follows from these considerations that, when we wish to consider what is happening in some very small region (as we have to do whenever we apply the differential or integral calculus), we must not take merely a small region of space, but a small region of space-time, i.e. in conventional language, what is happening in a small volume of space during a very short time. This leads us to consider, not merely the energy at an instant, but the effect of energy operating for a very short time; and this, as we saw, is of the nature of action (in the technical sense). A quotation from Eddington[13] will help to make the point clear:

“After mass and energy there is one physical quantity which plays a very fundamental part in modern physics, known as Action. Action here is a very technical term, and is not to be confused with Newton’s ‘Action and Reaction.’ In the relativity theory in particular this seems in many respects to be the most fundamental thing of all. The reason is not difficult to see. If we wish to speak of the continuous matter present at any particular point of space and time, we must use the term density. Density multiplied by volume in space gives us mass or, what appears to be the same thing, energy. But from our space-time point of view, a far more important thing is density multiplied by a four-dimensional volume of space and time; this is action. The multiplication by three dimensions gives mass or energy; and the fourth multiplication gives mass or energy multiplied by time. Action is thus mass multiplied by time, or energy multiplied by time, and is more fundamental than either.”

It is a fact which must be significant that action thus turns out to be fundamental both in relativity theory and in the theory of quanta. But as yet it is impossible to say what is the interpretation to be put upon this fact; we shall probably have to wait for some new and more fundamental way of stating the quantum theory.

There is one other respect in which some of the later developments of relativity suggest the possibility of answers to questions which have hitherto seemed quite unanswerable. Our theory, so far, leads us to brute facts which have to be merely accepted. We do not know why there are two kinds of electricity, or why opposite kinds attract each other while similar kinds repel each other. This dualism is one of the things which is intellectually unsatisfying about the present condition of physics. Another thing is the conflict between the discontinuous process by which energy is radiated from the atom into the surrounding medium, and the continuous process by which it is transmitted through the surrounding medium. Relativity throws very little light on these points, but there is another point upon which it throws at least a glimmer. We find it hard to rest content with the existence of unrelated absolute constants, such as Planck’s quantum and the size of an electron, which, so far as we can see, might just as easily have had any different magnitude. To the scientific mind, such facts are a challenge, leading to a search for some way of inter-relating them and making them seem less accidental. As regards the quantum, no plausible suggestion has yet been made. But as regards the size of an electron, Eddington makes some suggestive observations, which, however, require some preliminary explanations.

We saw that, according to the theory of relativity, the interval between two events may be separated into a time-part and a space-part in various ways, all of which are equally legitimate, and each of which will seem natural to an observer who is moving suitably. The first effect of this is to diminish the sharpness of the distinction between space and time. But the distinction comes back in a new form. It is found that the interval between two events can, in some cases, be regarded as merely a space-interval; this will happen if an observer who is moving suitably would regard them as simultaneous. Whenever this does not happen, the interval can be regarded as merely a time-interval; this will be the case when an observer could travel so as to be present at both events. It takes eight minutes for light to travel from the sun to the earth, and nothing can travel faster than light; therefore if we consider some event which happens on the earth at 12 noon, any event which happens on the sun between 11.52 a. m. and 12.8 p. m. could not have happened in the presence of anything which was present at the event on earth at 12 noon. Events happening on the sun during these 16 minutes have an interval from the event on earth which will, for a suitable observer, seem to be a spatial separation between simultaneous events; such intervals are called space-like. Events happening earlier or later than these 16 minutes will be separated from the event on earth at noon by an interval which would appear to be purely temporal to an observer who had spent the interval in travelling from the sun to the earth, or vice versa as the case may be; such intervals are called time-like. Two parts of one light-ray are on the borderland between time-like and space-like intervals, and in fact the interval between them is zero. But in all other cases there is a separation which is either time-like or space-like, and in this way we find that there is still a distinction between what is to be called temporal and what is to be called spatial, though the distinction is different from that of every-day life.

For reasons which we cannot go into, Einstein and others have suggested that the universe has a “curvature,” so that we could theoretically go all round it and come back to our starting-point, in the sort of way in which we go round the earth. All the way round the universe, in that case, must be a certain length, fixed in nature. Eddington suggests that some relation will probably be found between this, the greatest length in nature, and the radius of the electron, which is the least length in nature. As he humorously puts it: “An electron could never decide how large it ought to be unless there existed some length independent of itself for it to compare itself with.”

He goes on to make another application of this principle, which is suggestive, though perhaps not intended to be treated too solemnly. The curvature of the universe, if it exists, is only in space, not in time. This leads him to say:[14]

“By consideration of extension in time-like directions we obtain a confirmation of these views which is, I think, not entirely fantastic. We have said that an electron would not know how large it ought to be unless there existed independent lengths in space for it to measure itself against. Similarly it would not know how long it ought to exist unless there existed a length in time for it to measure itself against. But there is not radius of curvature in a time-like direction; so the electron does not know how long it ought to exist. Therefore it just goes on existing indefinitely.”

But even if the size of an electron should ultimately prove, in this way, to be related to the size of the universe, that would leave a number of unexplained brute facts, notably the quantum itself, which has so far defied all attempts to make it seem anything but accidental. It is possible that the desire for rational explanation may be carried too far. This is suggested by some remarks, also by Eddington, in his book, Space, Time and Gravitation (p. 200). The theory of relativity has shown that most of traditional dynamics, which was supposed to contain scientific laws, really consisted of conventions as to measurement, and was strictly analogous to the “great law” that there are always three feet to a yard. In particular, this applies to the conservation of energy. This makes it plausible to suppose that every apparent law of nature which strikes us as reasonable is not really a law of nature, but a concealed convention, plastered on to nature by our love of what we, in our arrogance, choose to consider rational. Eddington hints that a real law of nature is likely to stand out by the fact that it appears to us irrational, since in that case it is less likely that we have invented it to satisfy our intellectual taste. And from this point of view he inclines to the belief that the quantum-principle is the first real law of nature that has been discovered in physics.

This raises a somewhat important question: Is the world “rational,” i.e., such as to conform to our intellectual habits? Or is it “irrational,” i.e., not such as we should have made it if we had been in the position of the Creator? I do not propose to suggest an answer to this question.