It is evident, therefore, that the concepts which enter the field equations have entirely lost their large scale significance; they have become blurred, fused together, and fewer in number. A precise analysis of this situation has probably never been attempted and would obviously be difficult: it would be interesting to know at least how many really independent concepts there are at this order of phenomena. An attempt at an analysis would probably be worth while from a physical point of view in suggesting possible experiments by which the number of physically independent concepts could be extended.
Since the quantities in the field equations are meaningless in the naked form in which they enter the equations, it is meaningless to inquire whether the equations as they stand are true or not. In our present state of experimental knowledge it is also meaningless to ask whether, for example, the inverse square law between electric charges continues to hold, or whether an accelerated charge radiates. These questions have meaning only when applied to phenomena on a scale large enough to correspond to possible experiment.
There is a rather interesting obverse to the statement that it is meaningless to ask whether the field equations are true, namely, that it may not be meaningless to state that they are false. A statement is not true unless it is true in every particular, but it is false if it is false in a single particular. If we can show that a single consequence of the field equations of Lorentz, when integrated or averaged in such a way as to correspond to experimental possibilities, is false, then the equations must be false. It seems that, regarded as a complete description of physical behavior on a small scale, the equations must be judged false, because they contain no suggestion of quantum phenomena.
Even if we have to recognize that the equations are false, there can be no question that they correspond to an important part of reality, and that they have been of the greatest service to physics. What is the significance of the success that they have attained? It is to be noticed that all the phenomena to which the Lorentz equations have been successfully applied, although not large scale phenomena in the ordinary sense of the word, are nevertheless phenomena involving the coöperation of a number of atoms, and that the equations unquestionably fail when applied to phenomena involving single electrons. It appears from our best present evidence that on a small scale the behavior of nature is governed by quantum principles and is therefore quite different from large scale behavior, Which we have seen is governed by the Maxwell equations. There must of course be a transition zone in which the character of phenomena changes from quantum to Maxwell. Now any program like that of Lorentz is almost inevitably bound to begin to give correct results when we get up as far as the transition zone, for the simple reason that the relations of Maxwell have been put into the equations and are always there ready to appear as soon as the quantum relations begin to give way. The physical significance of the success of the Lorentz program seems to be that the transition from Maxwell to quantum takes place at a stage pretty far down toward the individual atoms. To find the precise details of the transition from Maxwell to quantum phenomena constitutes a large part of the program of the immediate future.
All this skepticism about the classical work of Lorentz is likely to be rather irritating or depressing, particularly if one attempts to imagine what other course could have been adopted. Indeed it does seem that we find ourselves in a real quandary; Lorentz was practically forced, because of the character of the mathematical tools at his command, to take the course that he did, in spite of any recognition of the physical meaninglessness of the mathematical operations. We have already seen that conventional mathematics does not correspond to the physical reality; it cannot easily make a qualified statement subject to limitations, and it recognizes no difference between the physically big and the physically little and the corresponding change in the operational meaning of its symbols. It begins by being a most useful servant when dealing with phenomena of the ordinary scale of magnitude, but ends by dragging us by the scruff of the neck willy nilly into the inside of the electron where it forces us to repeat meaningless gibberish. Larmor recognized this, and in his electron theory, developed practically contemporaneously with that of Lorentz, endeavored to treat electrons as wholes, and not to make meaningless statements about their insides.[25] But he was much less successful than Lorentz in making his analysis give physical results, and one may suspect that it was at least in part due to difficulty with his tools.
[25]Joseph Larmor, Æther and Matter, Cambridge University, Press, 1900. In this book the electron is treated as a point singularity in the ether.
What we should like to be able to do is easy to see. The things that go into our equations must have independent physical meaning, and the character of our mathematical formulation should change to keep pace with the change in the physical operations which give meaning to the terms. For example, electrical density has a meaning for large scale phenomena, but means nothing on a small scale. Our ultimate electric unit is the electron; when we get down to this scale of magnitude, our mathematics ought to be making statements about the relative behavior of discrete electrons, and not mention so much as by implication the density at points inside an electron. But this sort of thing we apparently cannot yet do; the proper mathematical language has not been developed. Such a language, when developed, must not only be able to resist the temptation to burrow inside the electron, but must also try to get along without the field concept, which we have seen is liable to so much physical abuse, and must reduce effects in complicated electrical systems to the ultimate elements that have physical meaning, namely, a dual action between pairs of electrical charges, with no implications about physical action where the charges are not.
THE NATURE OF LIGHT AND THE CONCEPTS OF
RELATIVITY
We have already discussed several aspects of the theory of relativity in connection with the relation to it of some of our fundamental concepts. There are still other topics connected with relativity which demand attention; most of these involve the properties of light. It will now be convenient to discuss together the properties of light and these concepts of relativity. We restrict our discussion of light to those simple properties which bear on the theory of relativity.
Practically all our thinking about optical phenomena is done in terms of an invention, by means of which these phenomena are assimilated to those of ordinary mechanical experience, and so made easier to think about. To realize that invention has been active here, we must think ourselves back into that naive frame of mind in which experience is given directly in terms of sensation. The most elementary examination of what light means in terms of direct experience shows that we never experience light itself, but our experience deals only with things lighted. This fundamental fact is never modified by the most complicated or refined physical experiments that have ever been devised; from the point of view of operations, light means nothing more than things lighted. Now experience shows that these things lighted may stand to each other in varied relations; in attempting to reduce these relations to order and understandability we make a certain invention. This is prompted by several cardinal experimental facts: in the first place, things lighted have a simple geometrical relation to each other, in that screens placed on straight lines between the lighted objects may suppress the illumination of one or the other and themselves become illuminated. This leads to the concept of rectilinear beams of light, which is no more than a description of the geometrical relation between lighted objects. Then we have the experimental fact of the asymmetrical relation of the lighted objects, described in terms of sources and sinks. Finally, we have the discovery made at a much later stage, and not possible until physical measurements had reached a high refinement, that light has properties analogous to the velocity of material things. This was first discovered in connection with astronomical phenomena in the shift of the time of eclipse of Jupiter's satellites and in aberration, but was later found to hold for purely terrestrial phenomena, in that a beam of light reflected from a distant mirror does not return to the source until after the lapse of a time interval that can be measured with means sufficiently refined. This property of return after the lapse of time is precisely like that of material things, such as a messenger despatched for an answer, or a ball or a water wave bouncing from a wall. These various properties of light lead quite naturally and almost inevitably to the invention of light as a thing that travels, "thing" not necessarily connoting material thing.