Part 1 — CHAPTER X CLASSICAL MECHANICS AND THE NEWTONIAN PRINCIPLE OF RELATIVITY

WE mentioned, when discussing amorphous mathematical space, that position and hence motion could have no meaning other than that of expressing conditions with respect to some particular frame of reference. In short, the visual principle of relativity would have to be adhered to.

We will accordingly select some arbitrary system of reference, which we will assume to be defined by three mutually perpendicular planes meeting at a common point . The instantaneous position of a body will henceforth be defined when we have measured with rigid rods its distances from the three planes of our frame; and the only type of position and of motion of the body which we can define in an unambiguous way will be its position and motion as referred to our frame.

The measurement of the velocity of the body with respect to our frame necessitates the addition of a clock. We will assume that this clock beats out congruent intervals of time, where congruence is, of course, defined by the requirements of practical congruence for time, that is, by the beatings of some isolated periodic mechanism.

Now, it is perfectly apparent that unless referred to some particular frame, the path followed by a body and its motion along this path can have no determinate meaning. Thus, if in a uniformly moving train we throw a ball into the air and catch it again in our hands, then, as referred to the train, the ball will have followed an up-and-down motion along the same vertical; on the other hand, as referred to the embankment, the ball will have described a parabola. Again, if a stone is allowed to fall from a great height, its motion with respect to the earth will be accelerated, whereas, with respect to a frame of reference falling together with the stone, it will have remained at rest.

It would thus appear that we might credit to the path followed by a body any arbitrary shape we wished; all we should have to do would be to select some appropriate frame of reference. The same indeterminateness would, of course, apply to the motion of the body along its path. Thus the shape of a path in space (rectilinear or curved) or the species of a motion (uniform or accelerated) could have no absolute significance. From the standpoint of mathematical space these would all be essentially relative to the reference frame we had selected.

But it is obvious that this attitude into which our understanding of mathematical space has forced us presupposes that all the frames of reference we might select were on exactly the same footing, that is, were indistinguishable from one another. Galileo and Newton noticed that such was not the case. In certain frames we were pulled hither and thither by strange forces unsymmetrically distributed, and objects placed on the floor did not remain where they had been put. Space around us appeared to be in a chaotic condition, as, for example, on a rotating disk or in a train rounding a curve. In other frames no such forces were apparent (excluding the force of gravitation), and space appeared stagnant and quiet, the same everywhere. These latter frames are called Galilean or Inertial.

Now, a Galilean frame did not manifest itself as unique. So far as the most delicate mechanical experiments could detect, there appeared to exist an indefinite number of such frames, all moving in respect to one another with various constant speeds along straight lines, without suffering any relative rotation during their motion. Mechanical experiments conducted in any one of these Galilean frames showed that free bodies followed straight lines with constant speeds (Newton’s law of inertia), and when referred to these frames, mechanical phenomena, including the planetary motions, were susceptible of being formulated by very simple laws. As viewed from Galilean frames, the other frames, the non-Galilean ones (those filled with strange forces), were found to be moving with accelerated or rotationary motions.

Mechanical experiments conducted in the non-Galilean frames proved that the disposition of the strange unsymmetrical so-called inertial forces bore a close relationship with the apparent motions of these frames as viewed from the Galilean ones. It was possible, therefore, for an experimenter situated in one of these strange frames to anticipate, without looking outside, exactly how his frame would appear to be moving when viewed by a Galilean observer (i.e., an observer in a Galilean frame), at least so far as acceleration and rotation were concerned. Thus, in a non-Galilean frame moving with respect to a Galilean frame along a straight line with some definite acceleration, the forces of inertia would be disposed in a parallel way, pulling in the direction opposite to that of the relative acceleration. In a non-Galilean frame rotating with respect to a Galilean observer, the disposition of the inertial forces would be much more complicated. There would be the centrifugal force pulling outwards and the so-called Coriolis forces pulling sideways.

As viewed from all these non-Galilean frames, the laws of mechanics would appear hopelessly complex. Free bodies would no longer appear to follow straight lines with constant speeds, but would describe capricious curves with varying velocities. Newton’s Law of Gravitation would no longer account for the planetary motions as observed from these frames, and a study of mechanics would present tremendous difficulties. In short, all the simple laws observed from Galilean frames would have to be compounded with the effects of these strange inertial forces when we wished to formulate laws controlling phenomena as viewed from the non-Galilean frames.

No surprise need be felt, therefore, when it is said that classical science selected the Galilean frames as standard frames. All the laws of mechanics and of physics were referred to them, and this, indeed, was the essence of Copernicus’ discovery. Incidentally, we see that we are in possession of another definition of a Galilean frame; for we may say that a Galilean frame is one with respect to which the laws of classical mechanics (law of inertia, Newton’s law, etc.) hold true.

We may also mention that the very possibility of there being definite laws of mechanics, such as the law of inertia, is contingent upon the existence of certain fundamental differences between the various frames. If all frames were identical, as the complete relativity of motion would demand, the law of inertia would express nothing at all. In every case we might make the motion of a body appear anything we wished by selecting our frame suitably; and if we had no means of distinguishing one frame from another the straight path of a body through empty space would convey no physical meaning.

We may now give still another definition of a Galilean frame. It follows from the law of inertia, according to which free bodies, when viewed from Galilean frames, will describe straight lines with constant speeds, that the stars, being presumably free bodies, will obey this law and, owing to their remoteness, appear to suffer very slight displacements even in the course of a century This fact led to a more easily obtainable definition of a Galilean or inertial frame, namely, a frame with respect to which the stars would appear fixed. This definition is not meant to supplant the one given previously of a frame in which no forces of inertia are experienced, for theoretically both definitions are equivalent. Its sole advantage is that it leads to a more accurate empirical determination of a Galilean frame and shows us immediately that our earth does not constitute such a frame, since the stars appear to circle round the Pole star. As viewed from a true Galilean frame, our earth would therefore be rotating round the polar axis every twenty-four hours.

Had we sought to discern the non-Galilean nature of the earth frame by endeavouring to detect traces of inertial, centrifugal and Coriolis forces, the task would have been more difficult owing to the minuteness of these forces. Nevertheless, the forces have been detected, and they account for the protuberance of the equator, the clockwise and anti-clockwise motions of cyclones, the rotation of the plane of Foucault’s pendulum, the directions of trade winds and of ocean currents, etc.

Now, thus far we have not attempted to draw any general conclusions from the existence of these various frames. We have merely stated that they undeniably exist, since they have been revealed by experiment, and that it is simpler to refer the laws of physics, and particularly those of mechanics, to Galilean or inertial frames. When it comes to interpreting the deeper significance of all these facts, various alternatives present themselves, and in this chapter we will discuss Newton’s solution at some length.

The problem to determine is, “Why do different types of frames exist when our understanding of amorphous mathematical space would seem to preclude any such differences?” Here we may make one of two assumptions. Either we may assume that the differences in conditions which exist in the various frames (differences which are exemplified by varied distributions in the lay of the inertial forces, or again in their total absence) are intrinsic, or else we may assume that they are due to foreign influences. Newton adopted the first alternative, Mach and Neumann the second.

If we follow Newton we must assume that the peculiarities existing in our frames are not to be attributed to causes existing beyond them. We may therefore assume that the inertial forces would be exactly the same were the balance of the universe to be annihilated, so that to all intents and purposes we might consider our frames isolated from all external influences. We assume, then, that the various distributions of the inertial forces manifesting themselves in our various frames must be attributed to the conditions of rest and of motion of these frames, not with respect to the remainder of the universe, which would be as good as annihilated, but with respect to those limited portions of the ether or of space in which the frames are situated.

Newton thus objectivises space and, owing to the symmetrical conditions which endure in Galilean frames, assumes these latter to be at rest in space, whereas the non-Galilean ones are assumed to be accelerated or rotating. If, then, we remember that a strict correspondence exists between the centrifugal and Coriolis distribution of forces in a non-Galilean frame, and its rate of rotation as viewed from a Galilean one, we may assume that the space in which a Galilean frame is at rest is also that in which a non-Galilean frame is accelerated or rotating. There is, then, but one space, an absolute space with respect to which motion and rest have a physical significance. Galilean frames are those which are non-accelerated; non-Galilean ones are those which are accelerated or rotating in this absolute space. This was Newton’s solution, and provided we accept his postulate of isolation there appears to be no way of escaping it.

Under the circumstances Newton’s law of inertia, according to which a free body follows a straight course with constant speed in relation to a Galilean frame, acquires a more determinate significance. The law now implies that this motion is being described in absolute space—in a space, therefore, that possesses a definite structure and which is thereby able to guide the body along its straight course. Had space been entirely amorphous and relative, as we assumed originally, there would have been nothing to guide the body; and even had the latter possessed enough intelligence to guide itself, there would have been nothing in amorphous space for it to direct itself by.

A number of philosophers attacked the Newtonian hypothesis of absolute motion and absolute space. For instance, some of them remarked that this so-called absolute space might well be assimilated to a bubble of ether floating and rotating in a more embracing space, this other in yet another, ad infinitum; so that Newton’s absolute rotation would be meaningless. Needless to say, arguments of this sort are totally irrelevant to Newton’s stand and to science generally.

Classical science, by absolute space, meant one with respect to which motion, or at any rate accelerated motion and rotation, appeared to manifest themselves dynamically; and experiment proved conclusively that such a space existed. There was no use, and in fact no sense, in wondering whether this absolute space should be conceived of as rotating in some other space; for even if a hypothesis of this sort were ever to be vindicated, the fact remained that rotation appeared to be absolute in a certain space regardless of whether or not this space was the super-space par excellence. This was all that was meant by the absoluteness of rotation so far as science was concerned. In short, as scientists clearly recognised, there was only one way to refute Newton’s absolute space and motion, and that was to deny his postulate of isolation. Such, indeed, was the course adopted by Mach.

If we follow Mach and Neumann and assume that the various internal dynamical conditions which distinguish the various frames are due to influences of an extrinsic nature, we shall no longer be able to subscribe to Newton’s conclusions; for we shall no longer be able to assume that were the entire outside universe to be annihilated, the internal conditions in our frames would endure as before. Henceforth, when ascribing the appearance of centrifugal and Coriolis forces to the rotation of the body, we should have to imply that this rotation was relative to a something in the universe that was not the empty space in which the frame was situated. The natural conclusion was that this rotation was relative to that other form of existence, namely, to matter. Neumann suggested a mysterious body called the “Alpha Body,” to whose dynamical influence he ascribed the rise of inertial forces; but, as Poincaré remarked, “Of this mysterious Alpha Body all we can ever know is its name.” A hypothesis such as Neumann’s, which is incapable of verification, reduces to a mere metaphysical speculation of no scientific value whatsoever.

Mach adopted a much more plausible alternative. He assumed that the rotation of the frame which appeared to accompany centrifugal and Coriolis forces was a rotation with respect to the remainder of the physical universe, hence to the totality of the matter of the universe. As the vastest agglomeration of matter in the universe is given by the totality of the star masses, we are led to Mach’s conclusions, namely, that the forces of inertia arise from a relative rotation or acceleration of the frame with respect to the universe of stars. Were the stars, then, to be annihilated, centrifugal force and all forces of inertia would vanish, a rotating mass of water would not splash over the sides of a bucket, and so forth. It is, of course, scarcely necessary to mention that the actual visibility of the stars or our mere ability to imagine their existence has nothing to do with the problem; it is solely the causal influences produced by their masses and relative acceleration which are relevant.

As a matter of fact, there is no very essential difference between the solutions of Mach and Newton. Even if we follow Mach, unless we are to believe in action at a distance, we must assume that the stellar influences can reach us only as a result of a propagation from place to place through space. We may perfectly well assume that as a result of this influence space receives a definite structure, so that in fine the physical reality of rotation may still be ascribed to a rotation in space. There is a difference, however; for with Newton this structure of space is intrinsic to space itself and has nothing to do with the contingent presence of matter generally, whereas if we follow Mach this structure of space must be conceived of as extrinsic and caused by the matter of the universe. In the absence of matter, the structure of space would automatically disappear and the various frames become indistinguishable.

As we can see, Mach’s views lead to a partial vindication of the visual or kinematical principle of relativity. But the vindication is only partial, since in all cases the relativity of acceleration and rotation exists only as between the totality of the stars and a body, and not between any two bodies taken arbitrarily, as in the visual principle.

Against Mach’s solution there is the argument that the stars are too remote to produce any such palpable effects as the bursting of a rotating flywheel or the splashing of a revolving mass of water over the sides of the bucket. But, still more important, it appeared impossible for classical science to account for these dynamical actions attributed to the stars. Nevertheless, with the tremendous advance in our understanding of nature which the general theory of relativity has given us, Mach’s ideas become highly probable, though not completely vindicated at the present time. Astronomical observations conducted on the globular clusters may eventually solve the problem.

Let us now return to Newton’s solution of the problem, since it constitutes the embodiment of classical mechanics. As we have seen, Newton, by positing his postulate of isolation, rendered inevitable a belief in some real absolute medium, space or the ether, from which rotation and acceleration would derive a real meaning. But here we must recall that mechanical experiments, though clearly differentiating a Galilean frame from a non-Galilean frame, were totally incapable of differentiating one Galilean frame from another. It appeared impossible to discover, therefore, which particular Galilean frame was to be considered at absolute rest in stagnant absolute space and which ones were to be regarded as moving through space with uniform translationary motions. In other words, when we confined ourselves to mechanical experiments, velocity through space was relative, the only velocity susceptible of mechanical detection being relative velocity, that is, velocity of one frame with respect to another. Acceleration and rotation, on the other hand, were absolute, and could be detected and measured mechanically without our having to appeal to a Galilean frame taken as term of comparison.

Were we to appeal to a Galilean frame in order to check up our computations, it would be a matter of complete indifference which particular Galilean frame we might select. As referred to any one of these Galilean frames, a body would manifest exactly the same acceleration and rate of rotation, whereas its velocity would, of course, vary with the Galilean frame to which it was referred. This is what is meant by the absolute character of acceleration in contradistinction to the relative character of velocity.

Of course, even an accelerated body could be made to appear at rest or to be moving with uniform motion if we selected a suitable frame. But the frame would have to be non-Galilean or accelerated in order to ensure this result, and we have seen that non-Galilean frames can never (at this stage, at least) be regarded as being at rest in space. It follows that the appearance of a body’s motion relatively to any of these non-Galilean frames could never be considered representative of the real motion of the body in space.

The Principle of Relativity of Classical Science merely summarises these discoveries by stating that it is impossible for an observer situated in a Galilean frame to ascertain by any mechanical experiment whether he is at rest in space or in a state of uniform translationary motion. A simple illustration will clarify the meaning of these statements.

Consider a train moving with constant speed along a straight course; obviously there is no trace of acceleration in the train’s motion, since no forces of inertia are felt in its interior. In common practice, we should speak of the train as having a definite speed of so many miles per second. But if we reflect we shall see that this constant speed is computed with respect to the earth’s surface, considered as defining a Galilean frame of reference. Were the earth suddenly to become invisible, we should have no physical means of determining the exact speed of the train through space. But suppose, now, that the train were slowing down, speeding up or rounding a curve. In this case, even if the earth were invisible, the train would still possess at each instant a definite acceleration and this acceleration would still be perceptible, and would be measured by the forces of inertia pulling and pushing the passengers about in the train. We see, then, that acceleration has an existence per se and that, in contrast to velocity, it is determinate even in the absence of a frame of reference. Now, it follows from the classical principle of relativity that, absolute velocity being meaningless (at least so far as mechanics is concerned), the only type of uniform translationary motion which can be detected by mechanical means reduces to relative uniform translationary motion of matter with respect to matter; that is, motion with respect to something observable. The magnitude of this relative motion depends essentially on the Galilean frame we may select.

Relative uniform translationary motion can, of course, be detected visually, but it can be detected also in a number of other ways. Thus the relative motion of a razor blade on a strop causes heat due to friction. The relative displacement of a knife between the two poles of an electromagnet produces induced currents which manifest themselves by rendering it difficult to cut through the lines of force of the field. The relative motion of an incandescent atom produces a shift in the spectral lines (Doppler effect), and it is thanks to this shift that astronomers are able to determine the radial velocities of the stars with respect to the earth. In other words, though absolute velocity can never be detected by mechanical experiments and is therefore meaningless in mechanics, yet relative velocity of matter with respect to matter can be detected in a number of ways; so that this relative type of velocity is the only one that has any physical significance, or at least any mechanical significance.

We have now to consider a number of difficulties that arise from Newton’s conception of absolute space. In the first place, although Newton had postulated absolute space and motion, the fact remained that of all the absolute motions, absolute acceleration and rotation were the only ones that had ever been detected, at least by mechanical means. It seemed strange, if absolute velocity were to have a meaning, as indeed it must if space were absolute, that it should be so obstinate in refusing to reveal itself. We could not hold that our inability to discover which one of the Galilean frames was truly at rest was due to the crudeness of our mechanical experiments; for the entire structure of Newtonian mechanics would have come crashing down had any such suggestion been proved correct.

The reason for this statement is easily understood. The fundamental law of Newtonian mechanics states that the force acting on a body is equal to the mass of the body (an invariant) multiplied by the acceleration. Now, the acceleration of a body as computed in any Galilean frame is always the same; for the various Galilean frames differ only in their velocity through absolute space, and a velocity added to or subtracted from an acceleration can never modify an acceleration. The mass of the body being an invariant according to classical mechanics, the force must also measure out the same in all Galilean frames. In other words, it was essential to classical science that the fundamental law of mechanics should remain the same in all Galilean frames, and similar conclusions would have to apply to all the mechanical laws. But then no mechanical experiment, however precise, executed in one Galilean frame or another, could ever yield varying results; so that absolute velocity could never be known. In short, if Newtonian science were to stand, the failure of mechanical experiments to detect the absolute velocity of our Galilean frame through space could never be attributed to their lack of refinement, since any positive results would have overthrown the entire structure of Newtonian dynamics.

Here, then, was an absolute velocity which was a reality since space was absolute, but which on no account could ever manifest itself!

It was, of course, possible to suggest that space had a dual structure, partly absolute and partly relative; absolute for rotation and acceleration, but relative for velocity. But this intrinsic duality in the nature of space and motion was hard to accept; it seemed as though space and motion should be one thing or the other: either entirely absolute or else entirely relative. Moreover, it was extremely difficult to conceive of a duality in the structure of space, so the entire situation was most mysterious.

There was, however, a possible way out of the difficulty, and that was to assert that absolute velocity was real, truly enough, and that it might perfectly well be detected provided we appealed to other types of physical experiment—to electrodynamic and optical experiments, for instance, and not solely to mechanical ones. But we shall see that precise experiment has since proved that these expectations were also doomed to disappointment; hence it appeared perfectly clear that no manner of experiment, mechanical, optical or electrodynamic, would ever reveal this elusive absolute velocity.

Even had optical and electromagnetic experiments succeeded in detecting absolute velocity, everything would not have been plain sailing. It would then always have been possible to assert that the absolute velocity detected in this way was not absolute velocity through space, but velocity through the ether floating in space. But we may point out that this last objection carries very little weight. For reasons which we shall explain in the next chapter, it does not appear legitimate to dissociate the ether from space. Had optical experiments detected velocity through the ether, we should have had to assume that they had also disclosed the long-sought-for absolute velocity through space; and the Newtonian belief in absolute space would have been considerably strengthened.

However, it is unnecessary to dwell on what might have happened, seeing that optical and electromagnetic experiments have failed to reveal the slightest trace of absolute velocity. Accordingly we are led to Einstein’s special principle of relativity, which states that not only mechanically, but in every way, absolute velocity through space must escape empirical detection. Under the circumstances, unless we wish to retain in science an absolute velocity which can never be detected and which is therefore any one’s guess, there is no alternative but to state that space together with its ether has a dual structure: relative for velocity, absolute for acceleration. Einstein appears to have thrown us back on the extremely difficult task of conceiving of this dual nature for space. But the solution of this particular problem was forthcoming as soon as Minkowski discovered space-time. With four-dimensional space-time the reason for the duality becomes apparent; for the absoluteness of space-time (as will be explained in a later chapter) accounts for the relativity of velocity and the absoluteness of acceleration.

Thus in Einstein’s theory, in its original form at least, velocity remains relative and acceleration remains absolute, as in classical science,[37] the only difference being that the relativity of velocity is now more thorough, applying as it does to all manner of experiments and not merely to mechanical ones. It is only when we consider the more speculative part of the theory, that pertaining to the form of the universe, that the complete relativisation of all motion in the Machian sense occurs. Space-time is then found to be no longer an absolute existing per se but to be conditioned entirely by the matter of the universe. But, as we have already had occasion to mention, this part of the theory is still highly speculative, and in order to avoid unnecessary confusion, we would strongly advise the reader to eliminate it entirely from his mind until such time as he has acquired a proper understanding of the special and of the first part of the general theory. Accordingly we will ignore it for the present; and we may state that in Einstein’s theory motion possesses a dual nature, just as was the case in classical science.