The logic of modern physics

by P. W. Bridgman

The statement that two exactly similar isolated systems, starting from the same initial conditions (including past history in the general idea of initial condition) will run through the same future course of events involves as a corollary that if differences develop in the behavior of two such apparently similar systems these differences are evidence of other previous differences. The thesis that this corresponds to experience may be called the thesis of essential connectivity and is perhaps the broadest we have: it is the thesis that differences between the behavior of systems do not occur isolated but are associated with other differences. It is essentially the same thesis as that already mentioned in connection with "explanation", namely that it is possible to correlate any of the phenomena of nature with other phenomena.

If now the connectivity or correlation between phenomena is of a special kind, we have a causal connection; namely, if whenever we arbitrarily impress event A on a system we find that event B, always occurs, whereas if we had not impressed A, B would not have occurred, then we say that A is the cause of B, and B the effect of A. By suitably choosing the event A, we may find the effect of any event of which the system is susceptible.

The relation between A and B is an unsymmetrical one, by the very nature of the definition, the cause being the arbitrary variable element, and the effect that which accompanies it. Furthermore, A may obviously be the cause of more than one event B, and may cause a whole train of events.

The causal concept analyzed in this way is not simple by any means. We do not have a simple event A causally connected with a simple event B, but the whole background of the system in which the events occur is included in the concept, and is a vital part of it. If the system, including its past history, were different, the nature of the relation between A and B might change entirely. The causality concept is therefore a relative one, in that it involves the whole system in which the events take place.

In practice we now take an exceedingly pregnant step and seek to extend the concept, and rid ourselves as much as possible of its relativity. It is a matter of experience that there are often a great number of systems in which A is the cause of B. In many cases the causal relation persists through such a very wide range of systems that we lose sight entirely of the system, and come to assume that we have an absolute causal connection between A and B. For instance, when I strike a bell, and hear the sound, the causal connection persists through such a great number of different kinds of system that I might think that here is an absolute causal connection. Such an absolute causal connection would mean that always under all circumstances, the striking of the bell is accompanied by a sound. But all conditions means only all those conditions covered by experiment. Thus in the case of the bell, all our experiments were made in the presence of the atmosphere. The causal connection between the striking of the bell and the sound should have been always recognized in principle as relative to the presence of the atmosphere. Indeed, later experiments in the absence of the atmosphere show that the atmosphere does play an essential part. Now as a matter of fact, the atmosphere is so comparatively easy to remove that we very readily include the atmosphere in the chain of causal connection. But if the atmosphere had been impossible to remove, like the old ether of space, our idea of the causal connections between the striking of the bell and its sound might have been quite different. In actual physical applications of the causality concept, the constant background which is maintained during all the variations by which the causal connection is established usually has to be inferred from the context.

It is a matter of perhaps universal experience that the event A is accompanied by not only one event, which is the effect of A by definition, but A entails a whole causal train of events. It seems to be a generalization from experience that the causally connected train of events started by A is a never ending train, provided the system is large enough. This is perhaps not necessary in the general case, but if the event A involves imparting external energy to the system, or the action of external force (momentum change), there can be no question.

That there is a causal train started by A is particularly evident if A and B are separated in space. Thus in the case of the bell, the impulse given to the air by the vibration of the bell is propagated through the air as an elastic wave, which thus constitutes the causal train of events. The phenomenon of propagation is characteristic of causal connections of a mechanical character, and is the justification for the introduction of the time concept in connection with the causality concept, where it now appears for the first time. It is evident that when a disturbance is propagated to a distant point, the effect follows the cause in time, as time is usually measured.

We extend this result, and usually think that the effect necessarily follows the cause. We now examine whether this is a necessary result of the causality concept. If we are to talk about the time of events at different places, we must have some way of setting clocks all over space. If this is done arbitrarily, there is no necessary connection between the local clock times of a cause and its effect, but nevertheless the causality concept involves a certain temporal relation even in this most general case. Suppose that event A takes place at point 1 and its effect, event B, at point 2. We station a confederate at 2 who sends a light signal (or any other sort of signal) to 1, as soon as the event B occurs at 2. Then it is a consequence of the nature of the causality concept that the signal cannot arrive at 1 before event A occurs. For if it did arrive before A, we should merely omit to perform A, which by hypothesis is arbitrary, and entirely in our control, and then our assumption would be violated that the system is such that the event B occurs only when A also occurs. The same argument shows a fortiori that if the effect B occurs at the same place as its cause A, it cannot precede it in time. I cannot see that the nature of the causality concept imposes any further restriction on the time of B. The restricted principle of relativity, however, in postulating that no signal can be propagated faster than a light signal, virtually makes a further assumption about the temporal connection of causally connected events, namely, that the event B at 2 cannot occur before the arrival at 2 of a light signal which started from 1 at the instant that A occurred at 1. For if B did occur earlier, we could use events A and B as a signaling code, thus violating our hypothesis.

There is thus a closest connection in time, when time is extended over space as the theory of relativity directs, between cause and effect, depending on their separation in space; from this arises the relativity concept of the causal cone, which in the four dimensional manifold of space-time divides the aggregate of all those events which may be causally related from the aggregate of those which are separated by such a small interval in time and such a large interval in space that communication by light signals and therefore a causal connection is not possible. Given now two events A and B which are related as cause to effect in one system of reference, then they must be causally related also in every other system of reference. For if they were not, we could by definition of causality suppress the event A in one of the systems in which the causal relation does not hold, and this, because of the nature of the concept of event, involves suppressing A in all the systems, thus violating our hypothesis of a causal connection in the original system. The concept of event involved in this argument will be examined later. It appears then, that the fundamental postulate of relativity (that the form of natural laws is the same in all reference systems) demands that the temporal order of events causally connected be the same in all reference systems.

The whole universe at this present moment is often supposed to be causally connected with all succeeding states. This means that if we could repeat experience, starting from the same initial conditions, the future course of events would always be found to be the same. The truth of this conviction can never be tested by direct experiment, but it is something at which we arrive by the usual physical process of successive approximation. It is difficult to formulate precisely what we mean by "present" state of the universe, and there is every reason to think that such a formulation is not unique, but the concept contains the necessary implication that none of the events constituting the "present" can be causally connected. The events in distant places which constitute the present must be separated by an interval of time less than time required by light to travel between the two places.