Einstein had attempted to render radical this relativity of inertia and motion even before discovering his cylindrical universe. He had solved the problem by postulating certain boundary conditions on the fringe of the nucleus of stars in infinite space. Thanks to these boundary conditions, the mass of a body would vanish at infinity, far from matter. But these solutions were abandoned because they appeared to conflict with the low star-velocities. With the cylindrical universe, however, all these difficulties were overcome, and for this reason, among others, it satisfies a natural philosophical craving. Inasmuch as all these considerations present an intimate connection with Mach’s mechanics, we will discuss the significance of Mach’s ideas at some length.
As is well known, on account of the existence of centrifugal force, of the inertial frame and of the rectilinear paths followed by free bodies according to the law of inertia, Newton felt himself compelled to postulate the existence of absolute space. Einstein summarises Newton’s attitude in the following lines:
“In order to be able to look upon the rotation of the system, at least formally, as something real, Newton objectivises space.... What is essential is merely that besides observable objects, another thing, which is not perceptible, must be looked upon as real, to enable acceleration or rotation to be looked upon as something real.”
The opposite stand was taken by Mach, who endeavoured to account for the dynamical manifestations of acceleration and rotation without appealing to that suprasensible entity, “absolute space.” Instead of holding, then, that centrifugal force was generated by a rotation in absolute space, he urged that we should attribute it to a rotation with respect to the material universe as a whole. Inasmuch as the vastest agglomeration of cosmic matter is given by the totality of the star-masses, it would follow that the dynamical effects would be generated by the rotation of an object relatively to the stars, or, what amounts to the same thing, by a rotation of the stars relatively to the object. In fact, these two different alternatives would represent but a difference in phraseology, since with absolute space banished, apart from stars and matter generally, there would exist no other terms of comparison.
To this rotation relatively to the star-masses would be due the protuberance of the equator, the splashing of water in Newton’s bucket, the bursting of a rapidly revolving flywheel, etc. In short, all those dynamical effects which in Newton’s opinion betrayed rotation in absolute space, are held by Mach to betray a rotation relative to the star-masses of the universe. Now, it should be apparent that Mach’s relativity of rotation entails more than a mere kinematical representation. As he is careful to specify, this active rôle of the stars is due not to the accidental circumstance that they happen to be visible, but to universal gravitational actions which would be generated between matter and matter when a state of relative acceleration or rotation was present. In this way, forces of inertia and centrifugal force are no longer to be attributed to the intrinsic structure of absolute space; they now become akin to gravitational forces. Were the stars invisible, no change in the dynamical manifestations would be observed, but if their masses were to be annihilated, if the stars ceased to exist (not merely ceased to shine), all the telltale dynamical effects would disappear; a flywheel would not burst, water would not splash, and so on. In much the same way, the elliptical orbits of the planets are due to the gravitational attraction of the sun’s mass; not, of course, to the accidental circumstance that the sun happens to be luminous rather than dark.
From these conclusions further important consequences follow; in particular, we must assume that the inertial mass of a body must be generated solely by the mutual actions existing between this body and the other bodies of the universe. We can understand the reason for this statement when we realise that were all the stars to be annihilated, all inertial forces would disappear; but it is precisely the existence of inertial force which is responsible for the effort required to set a body in motion or to arrest it when started. For this reason the disappearance of inertial forces would automatically entail that of inertia or mass. We may note the contrast with Newton’s views, for in classical science the inertial forces arose from the structure of absolute space itself, and mass was regarded as an intrinsic property of matter. In the present case, mass joins weight in being a relative; for just as a body has no weight in the absence of other bodies, so now we see that, according to Mach, it would also present no inertial mass if situated all alone in an otherwise empty world.
When we consider Mach’s mechanics, we see that its net result has been to identify Newton’s absolute space with the space defined by the stars; absolute space has become materialised, so to speak. We may obtain a more concrete representation of the new ideas by imagining a crisscross of threads extending between each individual star and all the others. Rotation, which develops dynamical effects, is, then, the rotation of an object with respect to this network of threads; and it is obviously a matter of indifference to suppose that the object is rotating among the threads and stars, or that threads and stars are rotating around the object. Thus the inertial frame is any frame in which the threads appear to be non-rotating.
Mach’s mechanics might seem to draw too much on our credulity. We can grant that the sun exerts a gravitational effect on the earth, but to assume that the bursting of a flywheel is due to the presence of the stellar universe would appear to be stretching matters too far. Nevertheless, it should be remembered that Newton’s introduction of that suprasensible entity, absolute space, was also extremely distasteful to a number of thinkers, so that it is a question of choosing the lesser of two evils.
These ideas were upheld by Mach for a number of years; and although they can lay claim to no special originality, they are always referred to under his name. Mach’s stand has been so consistently misrepresented by philosophers that we feel justified in warning the reader by mentioning some of the erroneous assertions that have been made with respect to it.
Two centuries ago, Berkeley had defended views which present a certain superficial similarity to those of Mach. He argued that if there were no stars, no permanent points of comparison in the heavens, we could not even imagine the earth’s rotation, hence this rotation would be deprived of all meaning. To this argument the Newtonian would reply that it was, after all, not impossible to imagine absolute rotation as a rotation in a rigid jelly-like substance called space. Furthermore, he would add that it was the dynamical facts, the appearance of unsymmetrically distributed forces, that seemed to confirm the absolute nature of rotation. Absolute rotation was, then, totally irrelevant to our perception of luminous reference points in the heavens, or even to our ability to imagine their existence. So far as the Newtonian was concerned, even were the stars to be annihilated entirely, it would still be quite easy to determine whether or not the earth was in rotation. A number of experiments, such as that of Foucault’s pendulum, the gyroscope, and, more generally, the presence of centrifugal forces on the earth’s surface, would demonstrate the existence of this absolute motion. As against these arguments, Berkeley’s stand offers no reply. Only when, with Mach, we maintain that were the stars to be annihilated, these telltale effects would also disappear, only when we conceive of the stars not merely in a visual capacity or as a support for the imagination, but as producing causal gravitational influences, can an argument be presented against Newton’s absolute space. In short, according to Mach, it would be the gravitational action of the totality of the material universe, including the stars, whether visible or dark, the nebulæ, our sun, the planets and moons, that were responsible for centrifugal force.