Chapter III · DETAILED CONSIDERATION OF VARIOUS CONCEPTS OF PHYSICS (continued 2)
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Examination will at once justify this view. Thus we find a great many systems which are adequately described in terms of two variables, pressure and temperature, in that a function of p and t can be found such that its differential equals dQ + dW. There are other systems in which the six components of stress and t completely fix the internal condition in the sense that they determine a dE. In other systems the specification of a magnetic field may be necessary, or an electric, or a gravitational field. No case is known which cannot be handled in terms of the action of external forces of the proper kind, but there is no general procedure, and the first law owes its generality to the exhaustive cataloging of special cases.
We may now return to the question left in abeyance above of the ambiguity in dQ + dW. In all the cases in which the specific variables can be found which define dE, dQ and dW also have meaning. Consider, for example, a gas, the internal condition of which may be characterized in terms of t and p. The mere fact that the internal condition can be specified in terms of two variables, one a mechanical variable, shows that the substance is mechanically homogeneous. Being mechanically homogeneous, we do not have the possibility of ambiguous values of dW varying with the scale of the measuring instruments, and in fact we know that dW = p dv. Similarly the gas being homogeneous and at rest as a whole allows unique values for dQ. Of course this cannot obscure the physical fact that even in such a gas, when we go to a small enough scale, we find inhomogeneities arising from the Brownian movement, etc. Practically our statement means that the inhomogeneities are so fine grained that over a very wide range of scale of the measuring instruments we find the same definite results. The same sort of considerations apply to more complicated systems. If dE is a complete differential in terms of t and six stress components, this means again that the body is homogeneous, its condition is determined by temperature and stress, which are the same throughout the body, and again there is no possible ambiguity from the scale of the instruments which measure dW and dQ. It seems in general, then, that if the body allows operations by which dE acquires meaning, at the same time dQ and dW are provided for. In working out this idea in full detail, some care must be given to the question of order of differentials. dQ, for unit time and unit volume, is strictly equal to k∇2t, where k is thermal conductivity, so that in determining dQ the second derivatives of temperature are involved.
If the body is obviously not homogeneous, it is still a matter of experience that it can be divided into small pieces, each of which are by themselves sufficiently homogeneous, and the first law in its usual form may be applied to each of the pieces.
Finally, we emphasize a fact already implicitly mentioned, namely, that no physical significance can be directly given to flow of heat, and there are no operations for measuring it. All we can measure are temperature distributions and rates of rise of temperature. As at present defined, a heat current is a pure invention, without physical reality, for any determined heat flow may always be modified by the addition of a solenoidal vector, with change in no measurable quantity. If someone states that throughout all space there is a uniform heat current of 106 cal./cm.2 sec. in the direction of Polaris, no disproof can be given, for such a stream is solenoidal, and as much heat flows out of every closed surface in unit time as flows in. Such a solenoidal flow is meaningless in terms of operations; we could give meaning to such a flow only in terms of some slight modification of the solenoidal condition introduced by the measuring instrument. In all ordinary conditions the flow of heat given by the simple relation q = k Grad t corresponds exactly to what our atomic pictures lead to expect in those cases where the details of the picture can be worked out. But there may be cases where it is advantageous to supplement the ordinary heat flow (= k Grad t) by the pure fiction of a solenoidal flow, because in this way it may be possible to account for new phenomena which appear when the solenoidal conditions are slightly departed from. Thus if in a conductor at uniform temperature carrying a steady electric current we say that a heat current is also flowing proportional to the electric current and therefore solenoidal, we may provide the possibility for a simple correlation of phenomena found under those more complicated conditions when an electric current flows in a conductor of non-uniform temperature in a magnetic field. If it should turn out that the heat current is uniquely determined by considerations of this character, then we have taken the first step away from the pure formalism which this sort of thing otherwise is in the direction of giving physical reality to the invention of "heat current."
There are other interesting questions of a fundamental thermodynamic character, such for example, as whether the entropy concept has any general significance apart from the scale of our measuring instruments, and what is the operational significance of applying thermodynamic concepts to radiation, but we shall not consider these questions here.
ELECTRICAL CONCEPTS
We now set ourselves the problem of finding the meaning of the various concepts in terms of which we describe the behavior of electrical systems, assuming that we understand what we mean by "electrical." We start with the simplest electrical systems, namely, those in which we deal with static phenomena on a large scale. In such systems there are independent physical operations by which we may find the magnitude of any charge, provided that it is effectively concentrated in a geometrical point. The measurements involved in these operations are measurements of ordinary mechanical forces; we assume that our knowledge of mechanics has already taught us how to make such measurements. An electrically charged body experiences forces, which may be measured by tying a string to it and pulling on the string with a spring balance hard enough to keep the body in equilibrium. Three charges are numerically equal if when each is placed at unit distance from another, in the absence of the third (or other charge), the forces are always the same. If furthermore the forces are of unit magnitude, the charges are defined as unit charge. Having obtained unit charge, we define the magnitude of any other charge as equal to the force which it experiences when placed at unit distance from unit charge. This of course is all very trite; the important thing for us is merely that magnitude of charge, or quantity of electricity, is an independent physical concept, and that unique operations exist for determining it. These operations presuppose the ability to perform certain operations of mechanics. Having now learned how to measure electrical quantities, we discover experimentally the inverse square law of force, and later arrive at the concept of the electric field. As we have seen, the field is an invention; here we shall use this concept only for the purpose for which the invention was made, and shall not involve ourselves in any of the implications of ascribing physical reality to the field. Notice that as long as we deal only with point charges we do not have to define field strength in terms of the limiting procedure of making the exploring charge smaller, for the limiting small charge is necessary only to avoid the reaction of the exploring charge on the positions of the charges which generate the field. All this again is trite; the important point is that the operations by which the inverse square law and the concept of the field are established presuppose that the charge is given as an independent concept, since the operations involve a knowledge of charges. The operations also involve the measurement of forces by the ordinary static procedure of mechanics with spring balances. With the means now at our command we establish one very important property of electric charges, namely that the total amount of charge on an isolated body of finite size is conserved, no matter how the charge is forced to rearrange itself by the motion of charges on adjacent bodies.
By procedures exactly like those outlined above, we may treat all the corresponding magnetic quantities; there is formal parallelism between the two sets of phenomena, but there is the physical difference that we have to realize a single magnetic pole by the device of using a very long slender magnet.
We now give our electrical system more freedom, in that we allow the charges to be in motion with respect to each other. Perhaps the most immediate question which we now have to ask is whether charge continues to be conserved when set in motion, or whether the total charge on an isolated body is a function of its velocity? To answer this question we must generalize the procedure by which we assigned a numerical value to a stationary charge. Perhaps the simplest way is to allow two unit charges each to move with constant velocity, remaining at unit distance apart, and measure with a spring balance the force required to keep them at constant distance apart. Now we immediately find that the force is altered under these conditions, so that our first impulse is to say that the charge is a function of the velocity. But as we experiment further, we find that the state of affairs is very complicated; the force between the two charges at any moment of their motion depends not only on the charges, their distance apart, and their velocities, but also on the angle between the line joining them and the direction of motion in the lines. Further experiment of other kinds yields other information; it requires a force to maintain a charge in uniform motion in a magnetic field, or to maintain a magnetic pole in motion in an electric field. A moving electric charge exerts a force on a stationary magnetic pole, so that by definition the moving charge is surrounded by a magnetic field, and similarly a moving magnetic pole is surrounded by an electric field. Returning to our two moving electric charges, we are impelled to ask whether, if all these complications are possible, the numerical constant (unity for static charges) in the inverse square law of force is a function of velocity as well as the magnitude of the charges themselves? If we broaden the question in this way, as we apparently must, our problem becomes indeterminate, for we are trying to answer two different questions with a single kind of measurement, namely of the force between moving charges. I have had no better luck on trying other methods of measurement. Apparently the operations do not exist by which unique meaning can be given to the question of whether the magnitude of a charge is a function of its velocity. On realizing this situation, we are at first embarrassed to know how to proceed, but we reflect that the embarrassment is not of our own making, but corresponds to a physical fact. The concept of charge as a unique and independent thing essentially pertains only to static systems. We may extend the concept to moving systems if we wish, as a matter of convenience to ourselves, but must recognize that such an extension is an invention of ours and not a reality of nature. Now we do make such an extension, and we make it in the simplest possible way, that is, we define the charge on an isolated body in motion as that which we should find on it if we reduced it to rest and made measurements according to the regular static procedure. That this is a convenient thing to do depends on the experimental result that the charge so found is independent of the way in which velocity is imparted to or removed from the body; in other words, whenever the body is reduced to rest, the same charge is always found on it.
Although this is pure definition on our part, it turns out to have a most simple and convenient connection with experimental facts which were discovered after the decision to treat a moving charge in this way was made; the discovery is of the atomic structure of electricity. If then we agree to call each elementary charge a constant independent of the velocity, the total charge on a body becomes merely proportional to the count of the total number of atomic charges on the body, which is certainly highly convenient and suggestive.
Having now fixed what we mean by the magnitude of a moving charge, we are ready to turn to the general problem of the behavior of any system of charged bodies in motion. For the present we consider only phenomena of the scale of everyday experience. The most general problem that has meaning here is to determine all measurable properties of the system in terms of those data which experiment shows can be arbitrarily specified. Now we have already emphasized that the electromagnetic field itself is an invention, and is never subject to direct observation. What we observe are material bodies, with or without charges (including eventually in this category electrons), their positions, motions, and the forces to which they are subject. The forces are to be measured according to definition in mechanical terms, either by the strains in members of a framework if the system is in equilibrium, or in terms of accelerations and masses if it is not in equilibrium. The electromagnetic field as such is not the final object of our calculations, but the calculation of it is only an intermediate auxiliary step, convenient to make because our mathematical formulation gives so simple a connection between electromagnetic field, charges, and mechanical action that the latter can be calculated at once in terms of the former. In fact the connection is so simple that in many cases we have come to regard our problem as solved if we can compute the electromagnetic field, overlooking the fact that the field has no immediate meaning in terms of experience.
Electromagnetic theory now presents us with a solution of the general problem; this solution is contained in the four-field equations of Maxwell, the constitutive equations, and those additional equations (quite often lost sight of) which give the forces exerted by the field on electric charges, or currents, or dielectrics. Let us inquire how we may set about testing the physical correctness of these equations. We may begin with one of the simplest possible tests, and inquire whether the equations are correct in stating that the force acting on a charge moving in an electric field is simply the product of the charge and the field strength. This, on the face of it, is a surprising statement. The field itself is affected by the motion of the charges which generate it, and it is natural to expect a converse effect. If, furthermore, we have sympathy with the medium point of view, it is easy to think that whatever it is in the medium that gets hold of a charge and exerts a force on it will find it harder to take hold when the charge is in motion.
In attempting to check our statement experimentally, the only additional complication, as compared with the static case which we have already checked, is afforded by the motion of the charge, for we have defined the magnitude of a charge in motion, so there is no difficulty here, and we may furthermore suppose that the field is generated by stationary charges, so that we need not trouble to inquire whether the procedure by which the field was originally defined is here applicable. The task of checking the equation then reduces to the simple physical task of measuring the force on the moving charge. How shall we do this? If the velocity is low, we may tie a string to the charge and measure the force with a spring balance (or its equivalent). But now an examination of the equations shows that in more complicated phenomena perceptible deviations from the static behavior are to be expected only at much higher velocities than can be attained by towing charges with a string and a spring balance, so that it is evidently necessary to check the simple equation for the force on a moving charge also at high velocity. Since at high velocity the spring balance method for measuring forces fails, we are driven to the only procedure that we have, namely a measurement in terms of the resultant acceleration, calculating the force by Newton's first law of mechanics. But this involves a knowledge of the mass of the moving body, which we recognize in general may be a function of the velocity. Now we have already seen, in discussing the concepts of mechanics, that the operations by which mechanical mass is defined cannot be carried out at high velocities, so that either the concept of mechanical mass becomes meaningless at high velocities, or we must adopt another definition. In attempting to give this new definition of mass at high velocities, we are driven to a result of special relativity theory, namely that all mass, mechanical or electrical, must be the same function of velocity. If now electrical mass can be found in terms of velocity, our immediate problem is solved and we shall be in a position to complete the experimental check of the equation. But as a matter of fact, in order to determine electrical mass, we have to use that equation which we are now engaged in trying to establish. Logically we have again the vicious circle, the physical significance of which is that independent operations do not exist for giving unique meaning to the concept of force on a charge at high velocity.
We seemed so close to our goal a minute ago; that we may allow ourselves to jump the logical chasm, and assume that the equation is correct. Electrical mass now becomes a definite function of velocity, mechanical mass the same function, and we are in a position to compare the actual acceleration received by a charge in a field with that calculated by the equation. Our conviction, on the basis of all experience up to the present, is that the two accelerations will be found to agree.
The equation then does somehow make correct connection with experience in that a consequence of the equation can be verified experimentally, in spite of the fact that as the equation stands it is meaningless, because the operations do not exist by which meaning can be given to the individual terms. At low velocities the equation really says what it seems to say, because the individual terms have meaning in terms of operations; and, what is more, what the equation says agrees with experiment. At high velocities the equation does not mean at all what appears on the surface; by itself it has no meaning; it has meaning only when considered as a member of a system of equations, and only in so far as the system of equations makes by implication statements about nature that have meaning in terms of operations that can be carried out physically. The individual terms of the equation of the system do not have meaning at high velocities, and in fact there are more terms than there are independent physical operations.
An exact analysis from the operational point of view of the significance of the equations at high velocities has perhaps never been made, and is not necessary for our immediate purpose. The discussion has brought out, however, that the number of physically independent concepts has been cut down by two at least, in that we have made purely formal definitions of the meaning of quantity of electricity, and of the force exerted by a field on a charge at high velocity. There is no reason to think that there is anything unique about this analysis, or that formal definitions might not have been given to other concepts than charge and force. We can only state that as far as physical content goes the equations have at least two degrees of freedom. It should then be possible to find quite different sorts of equations which agree equally well with experience. In particular, since we have seen that the force on a moving charge has no meaning in terms of independent operations, it should be possible by arbitrary definition to make this force any function of velocity that we please (of course reducing to the proper value at low velocities), and then to determine the other equations so that the entire group of equations is consistent with experiment. So far as I know, no one has tried to give such a modified set of equations, and indeed there is no particular reason why anyone should bother to do this, because the present equations are simple enough, and the modified equations, although perhaps differing greatly in appearance from the present ones, would have no advantage in any greater or different physical content.
But there is no reason to think that the present state of affairs will always continue. We have seen that the decrease in the number of concepts corresponds to our inability to measure as many sorts of physical things at high velocities as at low. Now it is the task of the future experimenter so to refine the possibilities of measurement at high velocities as to restore these two degrees of freedom. In particular, mass should be made measurable in mechanical terms at high velocities. When this restoration has been made, and all the quantities in our equations receive independent physical meaning, the significance of the equations in terms of operations will be quite altered, although the formal appearance will be unchanged. We must then be prepared to find, as always when we change the range of phenomena, that the equations in their present form do not correspond to the facts at all, and that one of the alternative forms allowed by our present two degrees of freedom is the correct form. But until the new experimental facts have been obtained, it seems hardly worth while to attempt to specify the doubly infinite variety of forms which the equations might have consistently with present experiment.[23]
So far we have discussed the extension of ordinary electric phenomena in only one direction, to high velocities. There is another extension which is much more important physically, namely to very small scales of magnitude. This extension is necessary to an understanding of the properties of matter in bulk, the electrical nature of the atom having been once established. Our problem is to show how the statistical average of the behavior of a large number of electrons gives the large scale effects which are within the reach of observation, and which are described by the equations we have just discussed. To get this statistical average we must be able to calculate at least certain features of the behavior of the individual electrons, which means that we must know the form of the equations down to dimensions of the order of those of an electron, or smaller. Now if one contrasts the scale of the supposed dimensions of the electron with the smallest dimensions on which we can make independent experimental verification of those equations, he must admit that there is an enormous chance for change in the type of equation beyond the limit that we can reach by direct experiment, and the chances of guessing the correct extension of the equation to small dimensions are correspondingly almost vanishingly small. (We may perhaps say that experiments on the Brownian motion on a scale a good many atoms in diameter bring us the closest possible directly, which means that we are 106 or 107 fold away from electronic dimensions.) In spite, however, of the apparently enormous chances against it, this program of extending the field equations to small dimensions and following out the consequences was exactly the program which Lorentz set himself.[24] That Lorentz saw that such a program might be carried through must be recognized as a vision of extraordinary genius, and that he was willing to devote to it the years of arduous and detailed calculation that he did is evidence of a pertinacity of purpose of the highest moral order.
We now have to examine critically this program and to inquire what is the significance of the measure of success that Lorentz attained. The precise extension of the equations that he made was very simple, for the large scale equations of Maxwell were taken over with as little change as possible. The equations are so familiar that it is not necessary for us to write them in detail; they express relations between the electric and magnetic force vectors (force and induction now becoming the same thing, the difference between them in ponderable bodies being one of the things that is to be explained in terms of the electrons), the space density of electric charge, its velocity, and the force acting on elementary charge. We have to notice that although formally the equations have changed little in appearance, nevertheless the physical content, as judged by the operations, has changed a great deal. Consider, for instance, the meaning of charge density. In the Maxwell equations, ρ was merely the number of discrete elementary charges per unit volume, the distances between these charges being supposed so small compared with the scale of the phenomena involved that their average effect could be fairly represented in terms of their numbers. In the Lorentz equations, on the other hand, ρ has a value different from 0 only inside the electron; everywhere else ρ = 0. Now an examination of the previous discussion, in which we questioned whether the magnitude of the charge might be a function of its velocity, will show that there are no physical operations whatever by which meaning can be given to ρ at individual points inside an electron. There is a single condition on this ρ, namely, that its integral throughout the total volume assigned to the electron shall equal the total static charge of the electron. Obviously a single scalar condition is a pretty blunt tool with which to attempt to determine a point function throughout a volume. Again, the equations talk about the velocity of the charge at interior points of the electron; what possible physical operations are there by which meaning can be assigned to the velocity of an amorphous structureless substance in regions inaccessible to experiment? Here again, the concept as a detailed description of the behavior at a point has become meaningless, and again there is a single integral condition, namely, that the v associated with every ρ must be such that when integrated over the volume of the electron it will give a total transport of charge equal to that carried by the electron in its motion. This again is a single condition on a function distributed through space. Still again, the equations contain the electric and magnetic vectors at points inside the electron. What is the possible meaning of these field vectors in terms of operations? Our procedure for finding the field at a point involves by definition finding the force on an electric charge placed at that point. But there is no charge smaller than an electron, and the procedure degenerates into a fiction. Again there is a single integral condition on the field vectors; the integral of the force on the assumed charge density when taken over the total volume of the electron must give a value corresponding to experiment. Except for this single condition, the concept of the field at points inside the electron is an invention without physical reality. Not only is the field concept meaningless at points inside the electron, but it is meaningless at points outside within a certain distance, because the exploring charge can never be made smaller than the electron itself, and so can never come closer than a certain distance.
The actual state of affairs is much worse than has already appeared. It was shown in the discussion of space and time that no independent physical meaning can be attached to lengths and times as small as have to be assumed in describing the behavior of the individual electrons. The operations Div, Curl, d/dt which enter the field equations are, therefore, physically meaningless as they stand; they have only a mathematical meaning which begins to acquire physical complexion in a most complicated way when the equations are integrated over large enough volumes.
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.
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.
The question now for us is whether we shall regard this as a mere invention, made for convenience in thinking, or shall go further and ascribe physical reality to it, that is, shall we think of light as capable of independent physical existence in the space between the matter that constitutes the source and the mirror? Now in spite of the resemblances pointed out above, there is at least one universal and fundamental difference between a thing that travels and light. We have independent physical evidence of the continued existence of the ball, for example, at all intermediate points of space; we can see it, or hear it, or feel the wind in the air as it passes, or even touch it. All these phenomena are independent of the initial and terminal phenomena, and hence by our criterion for the physical reality of an invention, we are justified in ascribing physical reality to the ball in transit. But with the beam of light it is entirely different; the only way by which we can obtain physical evidence of the intermediate existence of the beam is by interposing some sort of a screen, and this act destroys just that part of the beam whose existence we have thereby detected. There is no physical phenomenon, whatever by which light may be detected apart from the phenomena of the source and the sink (understanding that a mirror is included in the idea of, sink); that is, no phenomenon exists independent of the phenomenon which led us to the invention of a thing travelling. Hence from the point of view of operations it is meaningless or trivial to ascribe physical reality to light in intermediate space, and light as a thing travelling must be recognized to be a pure invention.
The status of light is exactly the same as that of an electric field; there is not the slightest warrant for ascribing physical reality to either at points of empty space—light and field-at-a-point have no meaning until we go there and make experiments with some material thing. Of course the electromagnetic theory of light makes this resemblance inevitable, provided the theory and our views of the nature of light and the field are correct.
It cannot be denied that there are some phenomena which uncritically considered appear to justify thinking of light as a thing that travels; these will now be discussed. Probably the argument to which most significance is usually ascribed is derived from the phenomena of energy. The passing of light from source to sink is accompanied by the transfer of energy. But energy is conserved, so that we have to ask where the energy is in the time interval between the emission of light from the source and its absorption by the sink. There is an obvious answer: the energy is in transit, of course, somewhere in the intermediate space between source and sink. If we think of light as propagated through a medium, then the medium is such that energy may reside in it, as in the electromagnetic theory of light, or if light is more material and ballistic in character, the thing that travels has itself energy. We notice in the first place that the conservation principle involves the time concept, because what we mean by conservation is that the total energy of the universe, at a fixed instant of time, is constant. That is, we have to integrate over all space the local energy at a definite instant of time, and this involves spreading the time concept over all space. It is further evident that unless we spread the time concept over space in the right way we shall not get conservation. The proof that it is possible to spread the time concept over space in such a way as to give conservation involves a knowledge of the properties of light. It would seem, then, that we ought not to assume conservation in deducing the properties of light, when a knowledge of the properties of light is necessary to establish conservation. These considerations cannot be accepted as final, however, until a detailed analysis has been made, and this would be most complicated. But there is a more important consideration derived from our previous critique of the energy concept, namely, that there is no basis for asserting that energy is localized in space at all; energy is not a physical thing, but rather what we would call a property of a system as a whole. If this view of energy be granted, the whole energy argument for light as a thing travelling, and also for the existence of a medium, falls. I believe that similar considerations apply to any arguments from the conservation of momentum.
The possibility of detecting light in apparently empty space by a screen constitutes perhaps the most immediate reason for considering light as a thing that travels. This point of view I believe is characteristic of the entire attitude of Einstein in deducing the theorems of the special theory of relativity. Einstein's light signal is for the purposes of the deduction thought of as a simple spherical wave spreading from the source and capable of being watched as it spreads by an observer outside the system, in much the same way that a water wave can be watched. Of course the light signal cannot actually be watched in transit, but we can come fairly close to this ideal by placing screens at any point we please to make the wave visible. It is true that the mere act of showing the existence of the light destroys that part of the beam whose existence is detected, but the screen needs only an infinitesimal amount of light to make it visible, and so by the usual physical argument we may suppose that the detecting screen produces only an infinitesimal modification of the total original light.
Our satisfaction with this picture evaporates if our present quantum views of the nature of light are correct. We can no longer think of the spherical light pulse as of irreducible simplicity, but it is an exceedingly complicated thing, perhaps more complicated than a gas from the point of view of kinetic theory, and simulates simplicity by some sort of averaging of the effects of the elementary quantum processes of which it is composed. If the principles of relativity are to continue to be regarded as fundamental, or even if they are to remain intelligible, we must apply our reasoning, not to spherical wavelets, but to the elementary process of which these wavelets are composed. Now the elementary quantum act is essentially a twofold thing: there is a discrete act of emission at some discrete material particle, and the act is consummated by another discrete act (absorption or scattering) at some other discrete particle. We cannot yet fully characterize the details of this twofold process, but have to connect the place at which absorption takes place with the place of emission by statistical considerations. It is evident, however, that to think of emission as starting some process like a spherical wavelet travelling like a thing through space presents an entirely incorrect view, because in the wave there is no hint of the discrete place which is to terminate it. We may say crudely that there is no way by which the wave can know what discrete material particle is to complete the emission process. We may perhaps try to save the situation by remembering that a spherical wave is polarized and so has a unique direction associated with it; but further examination shows that this does not help, because the unique direction is that of no energy flow, and absorption can take place in any direction except this. It appears then that instead of being a help, the thing travelling point of view is a positive embarrassment when we try to picture by means of it the essentially twofold nature of the elementary quantum act.
Another plausible argument for light as a thing travelling may be deduced from our principle of connectivity. Imagine a dark lantern with a shutter that can be opened or closed so as to emit a momentary flash of light, a distant mirror, and a receiving instrument near the source. One of the properties of light that we always assume is that no permanent trace of the act of emission is left behind in the source. The most minute examination of all the details of the lantern and its surroundings at some time after the emission of the flash has not yet shown any phenomenon that betrays a remembrance of the emission of the flash, unless perhaps we measure the total energy or momentum and have some way of knowing what the energy or momentum would have been if the flash had not been emitted, and in any event we cannot specify the moment in past time when the signal was emitted. In the same way we cannot tell from an examination of a mirror whether it has at any time in the past reflected a beam of light. Consider now two systems, each consisting of a source and a mirror distant 3 x 1010 cm., identical in all respects, except that in one a light signal was flashed from the source 1.5 seconds ago, and in the other only 0.5 seconds ago. According to our hypothesis, the most complete examination of source and mirror in either system fails to show the slightest difference, but nevertheless there is something essentially different about the two systems, for in one a light signal arrives at the screen in 0.5 seconds, and in the other not until 1.5 seconds. This violates what we have suggested might be regarded as the cardinal and most general principle of all physics, the principle of essential connectivity, which states that differences between two systems must be associated with other differences. A most obvious and simple way of maintaining our principle is merely to point out that the system really included more than we investigated: the system properly consists of source, mirror, screen, and all intermediate space, so that if we had examined intermediate space we should have found light there in transit in different positions in the two systems, thus correlating with the differences in subsequent history. This argument appeals to me as perhaps the strongest that can be advanced for the view of light as a thing travelling. But it seems in no way conclusive. Our principle of essential connectivity made no mention of the time concept, but we have somehow smuggled it in making the application above. We sought to give a complete description of our system at some one instant of time, and this involved spreading the time concept over space. This itself is a questionable operation and may be done in different ways. But, more important, what is the justification for supposing that the system can be completely described by giving a complete description of all the measurable parts of it at some one epoch? We have seen that in the most general case the principle of essential connectivity must recognize that the concept of "initial condition" of a system involves all the past history, and we may have here a case in point. The answer can be given only by experiment. In dealing with ordinary experience, when we do not have to distinguish between local and extended time, and are not dealing with optical phenomena, there can be little question that experience at least approximately justifies the expectation that future behavior is determined in terms of the present condition and that present condition may be specified in terms of the results of present operations performed in the system. But before extending this principle to phenomena in which we have to distinguish between local and extended time, we have to answer just that question which we are now considering; namely, whether there are physical phenomena taking place in apparently empty space, and whether therefore empty space has to be included in the system. We find ourselves again treading the vicious circle. Perhaps experience will show that the extension of the principle of connectivity to optical phenomena involves something like this: namely, the future at any point in a material system is determined by a complete description of the present state of the system in the immediate vicinity, and by a history of the behavior at more distant points, this history extending over longer and longer intervals of time as the point becomes more remote.
I believe, however, that these possibilities will not seem very satisfactory, and that most physicists will discover in themselves a very strong disposition to feel that the future is determined in terms of a complete description of some sort of instantaneous configuration, time being extended in some suitable way over space. This instinctive demand that the future be determined in terms of the present may easily be consistent with the optical phenomena in our two systems consisting of source, mirror, and screen, without involving the material existence of light in empty space, provided that our assumption that the emission of light leaves behind it no permanent record in the source was incorrect. It may be that detailed examination of a source after emission will disclose permanent traces, from which the instant of emission may be found by extrapolation. If the conviction of determinism of the future is strong, the physicist may well be impelled to search here for new phenomena indicating such a memory of emission.
Let us now inquire how our physical structure might be affected if we should give up the identification of light with a thing travelling. One consequence is that light need no longer be thought of as having the property of velocity, since velocity, in terms of immediate experience, is a property of things moving from place to place. Giving up the concept of light as a thing travelling would enable us, then, to adopt an alternative method of describing nature with a different concept of velocity; we have seen that it is possible to define velocity in terms of operations different from the usual ones, in such a way as to give the usual numerical results at small velocities, but different results at high velocities, and in particular to give an infinite velocity for light.[26]
There is now no objection to an infinite number associated with light, if we no longer think of light as having physical velocity. We may, if we like, continue for the sake of convenience to talk of the velocity of light, clearly understanding that the infinite value which must be ascribed to this velocity corresponds to the fact that the physical concept of velocity does not apply in all respects to light. We should now have to revise our process for extending the time concept over space, because this was formerly so done as to give light a finite velocity. We are now to make this velocity infinite, which is obviously to be done simply by setting a distant clock on zero at the instant it receives a light signal flashed from our clock at its zero. The behavior of material things now takes on a simple aspect—there is no longer a finite upper limit to the velocity that can be given a material thing, and light has no longer the paradoxical property of bearing the same finite relation to each of two material systems which differ from each other by a finite amount (that is, the first postulate of relativity that the velocity of light is 3 x 1010 in all reference systems). Light instead now bears the relation of infinitude to each of two systems which differ from each other by only a finite amount, and this is natural from the mathematical point of view.
However, all is not simplified by this change in the method of setting clocks, but a price has to be paid. The price is that we have to give up the simple connection between the velocity of a thing and its "go and come" time. Our changes have not affected local time; the time of passage of light to a distant mirror and return to the source is not changed, and is therefore still finite, although we describe the velocity of light as infinite. Now examination shows at once that there is no immediate connection between the concepts of "velocity" and of "go and come" time, because the operations involved are different. A measurement of a linear velocity according to our definition involves two clocks at two different places or else a clock travelling with the object, while "go and come" time demands only a single clock at a single place, and also involves necessarily a reversal of direction of motion in the object under measurement. We see then that, according to the definition adopted for velocity, we have the choice either of doing as Einstein did in the restricted theory of relativity and making "go and come" time very simply related to velocity; or we may say that refined physical measurements show that something of significance happens when the direction of motion is reversed, and that phenomena are not symmetrical with respect to a reversal of direction. The asymmetry which results from reversing the direction of motion we may visualize as a sort of curvature in space and time, as of a small piece of an arc of a circle bent back on itself, with the two ends diverging. This alternative way of treating velocity would mean that velocity can be measured simply only by a specially situated observer; this need not be considered disturbing, because in fact the operations have been defined only with respect to such an observer.
Which of these two possible treatments of velocity shall be adopted is to a certain extent a matter of convenience, determined by the sort of phenomena in which we are most interested and wish most to simplify. Einstein's chief concern was with optical phenomena, so that the motive for his choice is evident. In this choice of Einstein it is not very evident that the desire to make "go and come" time simply connected with velocity played a very prominent part, but it seems rather that the desire to think of light as a thing travelling, with a finite velocity, was much more influential. This way of thinking of light is fundamental to all the treatment of restricted relativity; without this sort of picture all the mathematical deductions would lose their simplicity and convincingness, for in all the deductions we inevitably think of ourselves as an observer from outside, watching a thing that we call light travelling back and forth like any physical thing.
Now there can be no doubt that, when choice is possible, convenience and simplicity are important considerations; but I believe that there is another much more important consideration, namely, the most perfect reproduction possible of the physical situation. It seems to me that it is very questionable whether Einstein, and all the rest of modern physics, for that matter, have not paid too high a price for simplicity and mathematical tractability in choosing to treat light as a thing that travels. Physically it is the essence of light that it is not a thing that travels, and in choosing to treat it as a thing that does, I do not see how we can expect to avoid the most serious difficulties. Of course the whole problem of the nature of light is now giving the most acute difficulty. The thing-travelling point of view, even as treated by Einstein, does not land us in a situation which is at all satisfying logically. We are familiar with only two kinds of thing travelling, a disturbance in a medium, and a ballistic thing like a projectile. But light is not like a disturbance in a medium, for otherwise we should find a different velocity when we move with respect to the medium, and no such phenomenon exists; neither is light like a projectile, because the velocity of light with respect to the observer is independent of the velocity of the source. On the other hand, in aberration we have a phenomenon similar to that shown by projectiles. The properties of light are more like those of a projectile than is perhaps commonly realized, as is shown in the papers of La Rosa[27] on the ballistic theory of light. The properties of light remain incongruous and inconsistent when we try to think of them in terms of material things.
Einstein's restricted relativity has made a great contribution in so grouping and coordinating the phenomena that they can all be embraced in a simple mathematical formula, but he does not seem to have presented them in such a light that they are simple or easy to grasp physically. The explanatory aspect is completely absent from Einstein's work.
In view of all our present difficulties it would seem that we ought at least to try to start over again from the beginning and devise concepts for the treatment of all optical phenomena which come closer to physical reality. No one realizes more vividly than I that this is a most difficult thing to do. If we are ever successful in carrying through such a modified treatment, it is evident that not only will the structure of most of our physics be altered, but in particular the formal approach to those phenomena now treated by relativity theory must be changed, and therefore the appearance of the entire theory altered. I believe that it is a very serious question whether we shall not ultimately see such a change, and whether Einstein's whole formal structure is not a more or less temporary affair.
Although it is exceedingly difficult to forsee what the treatment of the future will be like, it is easy to surmise certain of its features. In essence the elementary process of all radiation perceived as radiation is twofold. There is some process at the source and some accompanying process at the sink, and nothing else, as far as we have any physical evidence; furthermore, the elementary act is unsymmetrical, in that the source and the sink are physically differentiated from each other. This is the most complete expression of the physical facts; there is nowhere any physical evidence for the inclusion of a third element (the ether). Therefore all the phenomena apprehended by an observer (and this embraces all physical phenomena) can be determined only by the source and the sink and the relation to each other of source and sink, for there is nothing else that has physical meaning in terms of operations. This formula covers not only the possibility of such first order phenomena as aberration and the Döppler effect, but also shows that such second order effects as that looked for by Michelson and Morley must be non-existent. It will thus be seen that some of the consequences of relativity theory are implicitly contained in certain very broad points of view. One interesting question that must be answered before we can get very far with a new treatment is whether the elementary optical process is of necessity twofold, or whether we may have emission without absorption, that is, radiation into empty space. Lewis seems to imply in recent papers that this is not possible.[28] The astronomers have already pointed out difficulties in explaining phenomena like the temperature equilibrium of the planets if we suppose this is the case.
OTHER RELATIVITY CONCEPTS
We now turn to some of the other concepts of relativity. One of the most important of these is the "event"; in fact this concept is made fundamental by Whitehead.[29] We have already discussed the concept of "event" under the "identity" concept with which it is closely involved. The event is usually thought of by Einstein as merely an aggregate of four coordinates, three of space and one of time. The principle of general relativity, namely, that the laws of nature shall be of invariant form, when formulated mathematically, involves the assumption that nature may be analyzed into events, and is expressed by the requirement that the mathematical relations between the coordinates of a chain of events shall be invariant. The same idea is also expressed by Einstein in another form, namely, that nature may be completely characterized in terms of space-time coincidences. In elaborating this idea, Einstein assumes that the results of all measurement may be given in terms of such coincidences.
Now it appears to me most questionable whether the analysis of nature into events is possible or sufficient. With regard to the coincidence point of view, it seems perfectly obvious that the world of our immediate sensation cannot be described in terms of coincidences; how, for example, shall we describe in terms of space-time coincidence the photometric comparison of the intensity of two sources of illumination, or the comparison of the pitch of two sounds, or the location of a sound by the binaural effect?
To justify the coincidence point of view we apparently have to analyze down to the colorless elements beyond our sense perception. It does not seem unreasonable, perhaps, to expect that the universe is completely determined in terms of the positions as a function of time of all the positive and negative electrons; but to introduce such a thesis now certainly goes beyond present experimental warrant, and is contrary to the general spirit of relativity, which nowhere else involves any reference to the small scale structure of things. Even if we were willing to overlook all these objections, we would still have the fact that the difference between a positive and negative electron is not contained in any specification of the mere coordinates.
A further very important doubt in principle as to the possibility of the analysis of nature into events is afforded by the character of the concept of event itself. We have seen that the idea of event involves the existence of discontinuities, and that this can correspond only approximately to the physical fact, because discontinuities apparently lose their abruptness as we make our measurements more refined. The thesis that nature can be described in terms of discontinuities of a very small scale seems much too special to be made a fundamental part of a theory of the general pretensions of that of relativity. In fact this, as well as a consideration to be mentioned later, suggests that the argument and result of general relativity may be intrinsically restricted to large scale phenomena.
We now pass from these somewhat special questions to ask why it is that Einstein was able in the general theory of relativity to obtain new and physically correct results from general reasoning of an apparently purely mathematical character. We are convinced that purely mathematical reasoning never can yield physical results—that if anything physical comes out of mathematics it must have been put in another form. Our problem is to find where the physics got into the general theory.
There are two questions to be disentangled here: we have to consider in the first place the significance of the fact that Einstein has been able to describe relations in nature in mathematical form, and in the second place of the fact that he was able to arrive at the mathematical formulation of these physical relations by reasoning of apparently a purely mathematical character, from postulates of merely formal mathematical content (invariance of natural laws in generalized coordinates). Now the theory of relativity does not seem to differ in the first respect from any other branch of mathematical physics, such as the classical mathematical theory of electricity and magnetism, for instance, and this matter has already been touched in an earlier chapter. It is a fact that the behavior of nature can in many cases be expressed to a high degree of precision in mathematical language, and relativity is not unique in this respect. In any event, we must not allow this possibility of mathematical formulation to obscure the essential fact that all physical knowledge is by its nature only approximate, so that we may expect at any time to find, when we have carried our measurements to a higher degree of precision, that our mathematical expression of the laws was not quite exact, as seems now to be the case with Newton's law of gravitation, for example. I do not suppose that Einstein would claim that the statements of relativity differ in this respect from any of our other statements about nature, although apparently some of his followers see something more here. (From the operational viewpoint the meaning to be attached to "something more" is somewhat obscure.)
With respect to the second question, we may stop to notice that the special theory stands in quite a different position from the general theory. The special theory is much more physical throughout; its postulates are physical in character, and it is obvious that the physics got into the results through the postulates. It seems to me without question that Einstein showed the intuitive insight of a great genius in recognizing that there are mutual relations between physical phenomena which can be described in very much simplified language in terms of concepts slightly modified from those already in common use. In view of the remarks made on the nature of light, it is legitimate to wonder, however, whether the formulations of even the special theory will always stand. It seems to be true that all the facts of nature, even in the absence of a gravitational field, cannot be connected by the simple formulations of the special theory; that the physical relations are simple only in a sub-group; and that if we wish to deal with all optical phenomena, we have carried our simplifications too far, for the emission of a light signal is not a simple event, and light is not in nature like a thing travelling. Just the sorts of physical thing which are ignored in treating light as the special theory does are coming to be more and more important in the minds of physicists, and this is a reason for wondering whether ultimately Einstein's special theory may not be regarded merely as a very convenient way of tying together a large group of important physical phenomena, but not as being by any means a full or complete statement of natural relations.
With respect to the general theory, however, I believe the situation is quite different. The fundamental postulate that the laws of nature are of invariant form in all coördinate systems is highly mathematical, and of an entirely man-made character. Of what concern of nature's is it how man may choose to describe her phenomena, and how can we expect the limitations of our descriptive process to limit the thing described? Furthermore, Einstein's method of connecting his mathematical formulation and nature by way of coincidences of 4-events (three space, one time coordinates) seems to be very far removed from reality, since it entirely leaves out the descriptive background in terms only of which the 4-event takes on physical significance. Nevertheless, three definite conclusions about the physical universe have been taken out of the hat by the conjuror Einstein (shift of the perihelion of Mercury, displacement of apparent position of stars at the edge of the sun's disk, and the shift toward the infra-red of spectrum lines from a source in a gravitational field), and the problem for us as physicists is to discover by what process these results were obtained.
An examination of what Einstein actually did in deriving his results will show, I believe, that the situation is really different from that suggested above. In the first place, the requirement that the laws of nature be of invariant form actually places no restriction, as any one can see by setting himself the task of expressing, for example, an inverse cube law for gravitation in terms of generalized coordinates. The work of expressing such a law can be attacked in a perfectly routine way. (The essential difference between the invariability requirements of the special and general theories is to be noted; the special theory requires that the velocity of light, for instance, have the same numerical value in all allowed systems: the general theory merely that all laws have the same literal form, but with variable numerical coefficients.) But, as Einstein says, if any one actually attempts to carry through the work of expressing an inverse cube law in generalized coordinates, he will find the task prohibitively complicated, and will seek for some simpler formulation. What Einstein actually did, therefore, was to require that the laws of nature be simple in generalized form. Now we know that the law of gravitation as formerly expressed in ordinary coordinates as an inverse square law was approximately exact, and was also simple. Any deviations from this law are small, and all experience leads us to expect that to the first order of small quantities the deviations can be taken care of mathematically in the form of small correction terms to this law. This by itself gives nothing, however, because a small correction term can be added to our equations in an infinite number of ways. If, however, we know that the equation must be of a certain type after the correction terms have been added, the possibilities are so much restricted that the form of the correction term may be determined. In arguing as to the probable type of the equation, Einstein advanced the considerations by which physics gets into the situation.
In the first place, the special theory had prepared us for the possibility of finding that our measuring instruments might be modified in a gravitational field, analogously to the shortening of a meter stick when set into motion. In fact, special theory had indicated that in an accelerated system the modifications might be too complicated to be treated by that theory. In the absence, then, of specific information we must be prepared for the most general possible alteration in space-time in a gravitational field. In describing space-time we must therefore use coördinates adapted to handling the most general possible relations, and these are the generalized coördinates of Riemann, which had been already discussed by mathematicians. Going back now to Einstein's criterion that the equations are to be simple, we have the demand that the equations be simple in generalized coördinates, and of course they must also reduce to the ordinary equations (that is, the equations of special relativity) in space where there is no gravitational field. In deciding the further question as to what the type of equation probably is, we are influenced by considerations of convenience as well as by physical considerations. Practically the only type of equation that can be handled mathematically is linear, so that we shall certainly try first whether this type of equation may not continue to hold. Now the Newtonian law of the inverse square may be expressed in terms of a linear differential equation of the second order in the old Cartesian coördinates (Poisson's equation), so that our most immediate suggestion is that the equations remain linear and of the second order in generalized coördinates. As a matter of fact, this requirement turns out to be sufficient to determine the small correction term by which the ordinary equations can be generalized; Einstein's papers must be studied to see how this works out in detail.
All this looks pretty mathematical, but as a matter of fact there is much physical content, because systems which can be described by linear equations of the second order have definite physical properties. The requirement that the equations be linear corresponds to one of the most fundamental properties of our universe—the causality concept would not be possible or would be much modified in a universe governed by non-linear equations, for the joint effect of two causes acting together would not be the sum of their effects acting separately, so that the analysis of a situation into simple elements would be impossible and the causality concept probably would not have arisen. Furthermore, an equation of the type of Poisson of the second order means that there are propagation phenomena, and equations of mechanics of the second order involve the existence of a scalar energy function. If, then, the behavior of the universe can be described by differential equations at all, these equations must be linear of the second order if the universe is to have the broadest physical characteristics of our own universe. What Einstein really did, therefore, was to demand that even when space-time is warped by the presence of a gravitational field, those physical phenomena which can be described in terms of differential equations continue to be described by linear differential equations of the second order; that is, that nature continues to be describable in terms of a causality concept, with propagation phenomena, and a simple energy function. The consequences of a guess like this about the properties of nature appeal to our physical intuition as being worth following out, and of course we know the experimental justification.
Several general comments may now be made on the structure reached in this way. In the first place, the whole structure is only descriptive in character; we find certain correlations in nature which we describe with considerable completeness in mathematical equations, without introducing any new element of explanation or of mechanism. We have seen that as we increase our range from the realm of ordinary phenomena to phenomena of different character we arrive at a stage where for a time the process of explanation apparently halts, and we have to be satisfied with a statement of mere correlation between elements; later, however, these elements may be accepted as the ultimates in a broadened scheme of explanation, and the explanatory process resumed. Are we at such a stage now with the general relativity theory, and may later a new scheme of explanation be established based on the correlations of Einstein? This is of course a matter of individual judgment; I personally question whether the elements of Einstein's formulation, such as curvature of space-time, are closely enough connected with immediate physical experience ever to be accepted as an ultimate in a scheme of explanation, and I very much feel the need for a formulation in more intimate physical terms.
In the second place, we must repeat the comment already made in discussing time, namely, that there is still a very wide gap between the theory and its physical application, in that we have no way of identifying our physical clocks and our physical measures of time with the thing called time in the formulas. This gap must be filled by a specification of the physical structure of a clock.
It has always been very puzzling to understand why Einstein has so strenuously insisted that the shift toward the infra-red is an integral part of the general theory, and that if the shift is not found, the theory must fall. In other words, Einstein insists that the assumption that an atom is a clock is an integral part of his theory. I believe that this attitude may be due to a realization by Einstein of that very flaw in the logical structure which we are now emphasizing. In the absence of any method of specifying the details of construction of at least one clock, relativity becomes a purely academic affair, unless there exist in nature concrete things which may serve as clocks. Einstein must either be able to tell how to construct a clock, or else be able to point to a specific example of a clock. He chose the atom as the specific thing. Doubtless the reason was the apparent simplicity of the vibrating mechanism of an atom, as shown by the precise equality of the frequencies emitted by all atoms of the same element. If the atom is not a clock, where in nature can one be found? But in the last few years we have come to appreciate the exceedingly complicated quantum structure of an atom, and Einstein's thesis loses much of its instinctive appeal.
Since Einstein created the theory of relativity, it is perhaps ungracious to question his right to stipulate that the assumption that the atom is a clock is an integral part of the theory. This, however, degenerates to a mere matter of language, and does not touch the arbitrary nature of the procedure. It does not prevent us from having a second brand of relativity theory, that of X instead of Einstein, exactly like that of Einstein except that perhaps now the "clock" is constructed in terms of the life period of a radioactively disintegrating element. The only way to eliminate the arbitrariness seems to be to postulate that all natural processes, which run naturally of themselves independently of what we may do, may equally well serve as clocks and give the same results. But in answering the question of the operational meaning of "independently of what we may do" we shall effectively have to answer the question of what is a clock. This point of view may possibly, however, get us a little nearer to our goal of finding how to specify the structure of a clock.
Finally, the general theory is not completely general, but applies only to a certain range of phenomena, just as we saw that the special theory does not embrace all optical phenomena. The general theory applies only to those phenomena which can be described in terms of differential equations, that is, par excellence, to large scale phenomena. If quantum phenomena cannot be described by differential equations,[30] as apparently now they cannot, general relativity cannot by its very nature be applicable. General relativity does not give us a comprehensive formulation of the behavior of all nature, and as far as we can see, we are still as far as ever from such a general formulation.
ROTATIONAL MOTION AND RELATIVITY
Physically there is a great difference between the behavior of systems in uniform relative rectilinear motion and those in uniform relative rotation. The special theory of relativity states that there is a triply infinite number of systems with all possible uniform rectilinear velocities with respect to each other, in all of which physical phenomena have exactly the same mutual relations, that is, natural laws are the same. Now the mere formulation of the principle suggests the sense in which "system" is here used. It is obvious that "system" refers only to a part of the universe; we are not making a hopelessly academic statement about what would happen if we had an infinite number of universes to experiment with, but are talking about operations that may be approximately realized in our own universe. The "system" of the formulation we may think of as a completely equipped laboratory, out in empty space, so far from the heavenly bodies that they can have no effect. The different systems of the formulation are different laboratories, all built to exactly the same architectural blue prints. The phenomenon to which the postulates of relativity apply are phenomena which pertain entirely only to one or another of these laboratories. The meaning of this restriction is not completely definite and has, in any special case, to be judged partly by the context. Obviously, to see from the window of a laboratory another laboratory passing with a certain velocity cannot be counted as one of the allowed phenomena. Still less is it one of the allowed phenomena to observe that the center of gravity of the entire stellar universe has a certain velocity of translation with respect to the laboratory. The special principle of relativity contains by implication therefore the statement that certain very large and important classes of physical phenomena may be isolated and treated as taking place unaffected by the rest of the universe. Granted now the possibility of isolation, we have a second statement, which is usually treated as if it were the entire statement of the restricted principle, namely, that there is a triply infinite set of systems in which these phenomena run in the same way independent of the relative motion of the systems with respect to each other. When once the significance of the observation is grasped that absolute motion has no meaning in terms of operations, we see that this last statement takes immediately a most simple and satisfying aspect, in fact, so simple and inevitable that we are inclined to see in this the complete essence of the situation and regard the meaninglessness of absolute motion as affording peremptory proof of the restricted principle.
With this bias we now turn to examine the facts of rotary motion, and are disconcerted to find them quite different. No meaning in terms of measuring operations can be given to absolute rotary motion any more than to absolute translation, but nevertheless phenomena are obviously entirely different in different systems in relative rotary motion (phenomena of rupture, for example), so that apparently there are physical phenomena by which the concept of absolute rotary motion might be given a certain physical significance. Given two worlds like our own in empty space, but surrounded by impenetrable clouds, and each provided with a Foucault pendulum, then we believe that it is physically possible that we may find on one of these worlds the plane of rotation of the pendulum gradually changing in direction, while on the other it remains stationary. This difference we regard as possible without other accompanying physical phenomena which are causally related to the rotation of the pendulum (of course we have to make the two worlds of infinitely rigid material and eliminate other phenomena which we regard as purely incidental), so that we apparently have here a contradiction of our cardinal physical principle of essential connectivity. We are certainly not inclined to give up our principle, and we believe that as a physical fact, if the clouds could be evaporated, an observer in one world would find that he was rotating with respect to the system of the fixed stars, whereas the corresponding observer on the other world would find that he was stationary. Our principle of essential connectivity is therefore maintained, in that the rotation of the plane of the pendulum is connected with a rotation with respect to the rest of the universe of the entire world in which the pendulum is mounted. As far as I am aware, no other way of maintaining our principle has ever been suggested. But this demands that we give up our physical hypothesis of the possibility of isolating a system. There is here no question of limiting behavior; we believe that no matter how far our rotating world gets from the rest of the universe the Foucault pendulum would always behave in the same way; the system can never be isolated, but such local phenomena as the invariance of the plane of the pendulum are always essentially determined by the rest of the universe.
If now our system cannot be isolated, we must return to the phenomena of translational motion. In principle the act of isolation cannot be performed, the rest of the universe cannot be disregarded, and we should expect that different states of translational motion as well as different states of rotational motion with respect to the rest of the universe would have an effect on phenomena. We set ourselves the problem of understanding this apparent enormous difference between phenomena of translation and rotation. We remark that what apparently is a difference in principle may, in virtue of the approximate character of all measurement, be only a difference in magnitude, and that translational effects may exist too small to detect. A physical basis for such a difference may be found in the enormously different numerical values of translational and rotational velocities with respect to the rest of the universe attainable in practice. In describing phenomena of cosmic magnitude, we may plausibly measure the phenomena in units commensurable with the scale of the phenomena. Thus in measuring linear distances, we may perhaps choose as the unit of length the diameter of the stellar universe, and in measuring rotation, a complete reversal of direction with respect to the entire universe. This last means a change of angular orientation of 2 π, the first means a length of the order of 106 light years. Measured in such cosmic units the angular velocities attainable in practice are incomparably greater than linear velocities. We now see that it is possible that the real state of affairs is as follows: namely, phenomena in any system are affected by motion with respect to the entire universe, whether that motion is of translation or of rotation, and the magnitude of the effect is connected with the velocity of the motion by a factor which is of the general order of unity when velocity is measured in cosmic units. This last is merely an application of the argument so often made in physics as to the order of magnitude of unknown numerical factors, and will be found expanded on page 88 of my book on Dimensional Analysis. The linear velocities attainable in practice are now so exceedingly low that their effect has not yet been detected experimentally, but angular velocities are high, and the effect is easily demonstrable. In this light the special principle of relativity is no different in character from any other physical law; it is only approximate, and some day our measurements may become refined enough to detect its limitations.
We have made a hypothesis here, which we may call the hypothesis of the immanence of the entire universe, namely, that isolation is impossible, or that the rest of the universe, no matter how distant, always has a local effect on at least some phenomena. This is essentially the hypothesis of Mach,[31] and leads to a situation which can, I think, be contemplated with logical equanimity, although it has always seemed to many physicists most highly antiphysical in character.
It must certainly be admitted that most physical experience justifies us in thinking that effects may be made as small as we please by getting far enough away from the cause of the effect. But if we accept the considerations of the preceding pages, we must be prepared to admit that as phenomena change in range their character may change, and that in these new realms we must, at first at least, be satisfied with a mere statement of correlations. Certainly we have very strong physical evidence of a formal correlation between the Foucault pendulum and the rest of the universe. But a correlation of this sort may be without significance because of its very breadth; we never can prove the significance of the correlation by performing an experiment with the rest of the universe absent. Have we really done anything more than merely get things into such a formal situation that they cannot be assailed, a possibility which the mere laws of our thinking seem always to leave open, as has been suggested, or is there any physical content to what we have done? We have seen that if our correlation is also suggested by other phenomena, then we may accept it as having physical content. Now there is just a glimmer of a suggestion that our hypothesis of the immanence of the universe may be needed in other ways. The gravitational constant and the velocity of light are always treated as arbitrary magnitudes thrust on the universe from outside with no connection with other phenomena. Nevertheless, I suppose that no one regards this situation as ultimately satisfactory and does not entertain the hope that some day we may be able to give some sort of account of the numerical magnitude of these constants. We have not hitherto succeeded in finding any connection between these constants and small scale phenomena such as the charge on the electron, its mass, etc., so that there is some plausibility in expecting that a connection may be sometime found with cosmic things; indeed general relativity theory already prepares us for exactly this possibility. Now the velocity of light and the gravitational constant control small scale experiments, for of course these two constants can be measured by local experiments, so that if the cosmic connection is found, we should have a control of local behavior by cosmic things, and therefore another example of the immanence of the entire universe. There is no need for me to waste time in apologizing for the highly speculative character of all this. It is worth while to emphasize, however, that our general considerations on the meaning of "explanation" have prepared us to admit as reasonable just the sort of explanation contained in the hypothesis of the immanence of the universe, and therefore to reserve a place in our physical thinking for possibilities of this sort, in spite of the fact that such considerations are not usually entertained, and may seem to many opposed to the spirit of physics.
QUANTUM CONCEPTS[32]
The history of quantum theory up to the present is a repetition in many respects of that of the early theories of electricity, in that all our thinking has been in mechanical terms. As far as we now know, quantum phenomena are always associated with atoms. We make for the atom a mental model with all the properties of the mechanisms of the ordinary scale of magnitude and with a few impressed properties in addition which represent the new quantum relations. As we now think of it, the atom has a massive core about which electrons revolve under an inverse square law, the connection between the mass of the electron, its acceleration, and the force acting on it being that usual in Newtonian mechanics.
The space in which the electron circulates is thought of as Euclidean, and the motion is described in time, which may be measured with clocks in the usual way. The general equations of electrodynamics do not apply; there are no propagation effects inside the atom, the motion of the electrons does not produce a magnetic field, and there is no radiation when the electron is in one of its possible stable states, in spite of the acceleration. We may, if we please, in working out the character of the motion, entirely neglect the electrical origin of the inverse square law, and treat this merely as an impressed force without further implications. Superposed on the ordinary spatial, temporal, and mechanical characteristics of the model are additional quantum properties, one which determines the particular orbit in which the electron moves [∫ pdq = nh], and another which determines the frequency of the radiation emitted when the electron passes from one allowed orbit to another. No mechanism is suggested to account for these quantum conditions, although the conditions are formulated in mechanical terms.
We now have to ask what is the meaning in terms of operations of our usual concepts of space-time and mechanics when applied to phenomena of this order. It is of course evident, as has already been emphasized, that the concepts have entirely changed in character, because we do not measure an electron orbit, for example, by stepping off the diameter with meter sticks, or by measuring the time required for light to travel across the diameter. The particular feature of immediate interest in this changed situation is the change in number of our concepts on the atomic level. I shall not attempt to find by an exact analysis the number of independent concepts at this level; probably such an analysis is not possible. We may, however, make an approximate suggestion. Apparently the most important concept in describing relations inside a quantum system corresponds to that of energy on the ordinary scale. Changes of energy determine the frequency of emitted radiation, as well as the relations during collisions of atoms and electrons; these collisional relations make direct connection with experiment through the voltages applied to electrons in collision experiments. The analogue of the momentum concept also seems to have independent significance, as shown by the Compton effect. The frequency of emitted radiation is also something with independent experimental significance. I believe that these three things are all that have direct significance for quantum experiments made up to the present time. In any event, it is perfectly evident that on the quantum level the concepts which at present have operational significance are considerably fewer than on the level of ordinary experience.
Apart from the question of convenience, there may be justification in continuing to use our old mechanical forms of thought if new experimental relations are thereby suggested. That a very large number of such as yet undiscovered relations may be suggested in some such way is at once evident. Thus we have no present knowledge of any phenomenon associated with what the electron does when passing from one energy level to another. How long does it take to make the passage? What is its path during passage? Is it subject to the ordinary laws of electrodynamics during passage? When and where is the radiation emitted that corresponds to passage? When the electron leaves one stable orbit is the orbit on which it will eventually land already determined? Does the radiation train emitted during a change from one energy level to another have a definite length in space, or may it have a variable length and correspondingly something that corresponds to variable amplitude? What happens to the radiation when the electron passages are interfered with before the emission of a quantum has been completed? What is the mechanism by which the quantum conditions are imposed? Is it not possible that part of the clew to the riddle of the manner of transition from purely quantum behavior to the behavior of classical mechanics may be found in the behavior of the electron during passage from one energy level to another? Certainly we have a tendency to the classical behavior under those conditions, such as at high temperature or in strongly condensed systems, in which the time occupied in passage might be expected to become a more important part of the total time.