The logic of modern physics

by P. W. Bridgman

[16]Yale University Press, 1922.

If either of these two alternative definitions of velocity were adopted, it would be found that the velocity of light is infinite. Further, there would be no limit to the velocity which can be imparted to material bodies on giving them unlimited energy, which is what we are prepared to regard from ordinary experience is natural and simple. The infinite velocity for light, on the other hand, is most unnatural, particularly if we favor a medium point of view. We are here faced with a dilemma—all sorts of phenomena cannot at the same time be treated simply. If we attach the most fundamental significance to the behavior of material bodies, we shall do well to adopt one of the alternative definitions of velocity. If, on the other hand, we regard the phenomena of light as the most fundamental, we shall endeavor to form our definition so that the properties of light are simple. This was precisely the point of view of Einstein; it is characteristic of his entire scheme of restricted relativity that light is the fundamental thing, and this influenced him in adopting the first definition of velocity. Now one can have no quarrel with this desire to make light fundamental (the wisdom of doing this is to be justified by the results), and if the properties of light are to be treated mathematically, one can easily see the desirability of getting rid of infinite attributes, and so admit the desirability of making the velocity of light finite. But all this involves another very insidious assumption which we ourselves have tacitly used in all our preceding discussion, namely, that the notion of velocity properly pertains to light at all. Einstein has very definitely adopted this point of view, and so determined the character of the entire structure of relativity. I believe, on the contrary, that it is very gravely to be questioned whether the identification of light with a thing travelling, which is involved in applying the velocity concept, should be made. This discussion must be postponed, however, until we deal with the properties of light. The important points for us to notice at present are that the definition of velocity actually used involves the concept of extended time, and that it would be possible to define velocity in different ways, which would give quite a different complexion to phenomena at high velocities, but which would leave untouched our ordinary experience.

The velocities at which the precise form of definition becomes important are higher than can be reached in ordinary mechanical experiments. Such velocities can be attained in terrestrial laboratories only with electrified particles, as in experiments in high vacua or with radioactive disintegrations. It is interesting to notice that we very seldom attempt a direct measurement of velocity in such experiments by following a discrete particle in its flight and finding the time required to pass over a measured distance, but the velocities are measured indirectly, by calculation from the equations of electrodynamics and in terms of such immediately observed things as curvature of path. It is true that one or two experiments have attempted a more direct measure of velocity, but it seems there is room for more work here.

THE CONCEPTS OF FORCE AND MASS

Another concept of great importance is that of force. Since the usual analysis finds a connection between force and acceleration, and acceleration involves velocity, this is a natural place for the discussion of force. This concept has been subjected to much analysis by various writers. In origin the concept doubtless arises from the muscular sensations of resistance experienced from external bodies. This crude concept may at once be put on a quantitative basis by substituting a spring balance for our muscles, or instead of the spring balance we may use any elastic body, and measure the force exerted by it in terms of its deformation. Of course, the various precautions which must be taken in carrying out this idea physically are complex; the matter of precautions against temperature changes, for example, is one of the most easily understood. The concept of force so defined is limited to static systems; it is the task of statics to find the relation between the forces in systems at rest. We next extend the force concept to systems not in equilibrium, in which there are accelerations, and we must conceive that at first all our experiments are made in an isolated laboratory far out in empty space, where there is no gravitational field. We here encounter a new concept, that of mass, which as it is originally met is entangled with the force concept, but may later be disentangled by a process of successive approximations. The details of the various steps in the process of approximation are very instructive as typical of all methods in physics, but need not be elaborated here. Suffice it to say that we are eventually able to give to each rigid material body a numerical tag characteristic of the body, such that the product of this number and the acceleration it receives under the action of any given force applied to it by a spring balance is numerically equal to the force, the force being defined, except for a correction, in terms of the deformation of the balance, exactly as it was in the static case. In particularly, the relation found between mass, force, and acceleration applies to the spring balance itself by which the force is applied, so that a correction has to be applied for a diminution of the force exerted by the balance arising from its own acceleration.

We now extend the scope of our measurements by bringing our laboratory into the gravitational field of the earth, and immediately our experience is extended, in that we continually see bodies accelerated with no spring balance (that is, no force) acting on them. We extend the concept of force, and say that any body accelerated is acted on by a force, and the magnitude of this force is defined as that which would have been necessary to produce in the same body the same acceleration with a spring balance in empty space. There is physical justification for this extension in that we find we can remove the acceleration which a body acquires in a gravitational field by exerting on it with a spring balance a force of exactly the specified amount in the opposite direction. This extended idea of force may also be applied to systems in which there are electrical actions.

We thus see that in extending the notion of force from bodies in rest to bodies in motion, the character of the concept has changed, because the operations by which force is measured change—the force acting on a body is now measured in terms of its acceleration. But in determining the force from the acceleration, we have to know the mass. This mass has to be independently measured with the original concept of force; otherwise we have no basis for such simple statements as that the force of gravity on a body is proportional to its mass. All this applies to the ordinary range of experiments with low velocities. If now we extend the range of measurements, we find phenomena which we had not expected; for example, there seem to be difficulties in the way of indefinitely increasing the velocity of a material body, as of a charged atom. We begin to ask searching questions: is the force of gravity independent of velocity at high velocities, or is the mass independent of velocity under the same conditions or independent of the gravitational field, etc.?

In attempting to answer these new questions, we find difficulty with the concepts in terms of which they are formulated. There are no operations by which we can find whether force is independent of velocity unless we first know the mass, or any operations by which a mass can be measured unless we know a force. The purely mechanical systems with the highest velocities of which we have any experimental knowledge are the heavenly bodies. The motion of these is, with the important exception of Mercury, that predicted by the ordinary laws of mechanics, so that at first it might appear that we have here confirmation of the laws of mechanics for bodies with comparatively high velocities. But it must be remembered that all we can observe of the heavenly bodies is their positions, and that we cannot perform on these bodies all the operations by which we can check the laws of mechanics for terrestrial phenomena. If, for example, mass and the force with which gravity acts on mass were both equally affected by velocity, the motion of the heavenly bodies would be exactly the same as that observed now. Hence as we increase the range of velocity, the concepts of force and mass simultaneously lose their definiteness, and become partially fused. This is typical of what we have now come always to expect near the limit of the experimentally attainable; experience becomes less rich, the choice of physical operations more restricted, concepts change and become fewer in number. If we are to retain the same formal number of concepts, we must introduce arbitrary conventions or definitions. These definitions are to be determined largely by convenience. In the case of mechanical systems, this motive of convenience is supplied by considerations from outside the domain of mechanical phenomena. The highest velocities of practice are not reached in mechanical, but in electrical systems, in experiments with vacuum tubes, etc. Considerations of convenience are therefore dictated from the electrical point of view. These considerations will be gone into in much more detail later; the conclusion is all that we need here, which is that it is convenient to assume for the charge of the electron a constant number, independent of the velocity, and this involves making its mass variable in a definite way with velocity. Now if the principle of relativity is accepted, the mass of mechanical objects must vary with velocity in the same way as the mass of electrical charges. Since the variability of this latter is fixed, mechanical mass becomes a definite function of velocity, and the force is therefore also fixed in any specific physical case.

The fundamental definition of force given above is highly academic, involving as it does hypothetical experiments in laboratories situated far out in empty space. Some sort of procedure like this seems to correspond to more or less explicit statements to be found in the literature of mechanics. The meaning in terms of actual operations to be given to such definitions involves complicated inferential reasoning. We would make much closer connection with the conditions of actual experiment if in the definition we substituted for the hypothetical operations in empty space more or less approximately realizable operations on bodies sliding on level table tops without friction. I suppose our instinctive feeling for the laws of mechanics is such that we are convinced that definitions in terms of an interstellar space laboratory or a level table top are actually the same. But in principle we must recognize that when the operations are different, the concepts are different, and if we adopt something equivalent to the table top definition, as it seems we are physically forced to do, we must leave open in our thinking the possibility of finding in the present penumbra, when our accuracy is sufficiently increased, such phenomena perhaps as directional attributes of mass in a gravitational field.

We have just considered the sort of problem that we encounter on ordinary scales of magnitude on going from low to high velocities; what becomes of the concepts of force and mass when we go to a very small scale? Down to the atomic scale we may at least slur over the new physical difficulties, for although we cannot of course experiment with actual atoms, we can nevertheless make measurements of the Brownian[17] movement of suspensions in liquids settling in a gravitational field, for example, and the extrapolation to the atom is not a very great one. The mass of each individual atom is obtained by what is equivalent to a process of counting, assuming the law of conservation of mass on an atomic scale. This is justified by all chemical experience. The mass of the component parts of the atoms, the electrons, may also perhaps be given a unique significance after we have decided on the laws of the electrical field, by experiments on acceleration in electrical fields. The question which interests in principle here is what meaning, if any, shall be attached to the mass of the elements of the electron.