THE foundations of classical mechanics cannot be exhaustively described in a narrow space. I can only bring the unfavourable side of the theory into prominent view for the present purpose, without being able to do justice to its great achievements in the past. All doubts about classical mechanics set in at the very commencement with the formulation of the law which Newton places at its head, the formulation of the law of inertia.
As has already been emphasized on page 31, the assertion that a point-mass which is left to itself moves with uniform velocity in a straight line, omits all reference to a definite co-ordinate system. An insurmountable difficulty here arises: Nature gives us actually no co-ordinate system, with reference to which a uniform rectilinear motion would be possible. For as soon as we connect a co-ordinate system with any body such as the earth, sun, or any other body—and this alone gives it a physical meaning—the first condition of the law of inertia (viz. freedom from external influences) is no longer fulfilled, on account of the mutual gravitational effects of the bodies. One must accordingly either assign to the motion of the body a meaning in itself, i.e. grant the existence of motions relative to "absolute" space, or have recourse to mental experiments by following the example of C. Neumann and introducing a hypothetical body , relative to which a system of axes is defined, and with reference to which the law of inertia is to hold (Inertial system, vide Note 15). The alternatives with which one is faced are highly unsatisfactory. The introduction of absolute space gives rise to the oft-discussed conceptual difficulties which have gnawed at the foundations of Newton's mechanics. The introduction of the system of reference certainly takes the relativity of motions so far into account, that all systems in uniform motion relative to an -system are established as equivalent from the very outset, but we can affirm with certainty that there is no such thing as a visible -system, and that we shall never succeed in arriving at a final determination of such a system. (It will, at most, be possible, by progressively taking account of the influences of constellations upon the solar system and upon one another, to approximate to a system of co-ordinates, which could play the part of such an inertial system with a sufficient degree of accuracy.) As a result of this objection, the founder of the view himself, C. Neumann, admits that it will always be somewhat unsatisfactory and enigmatical, and that mechanics, based on this principle, would indeed be a very peculiar theory.
It therefore seems quite natural that E. Mach (vide
Note 16) should be led to propose that the law of
inertia be so formulated that its relations to the stellar bodies are
directly apparent. "Instead of saying that the direction and speed of a
mass remains constant in space, we can make use of the expression
that the mean acceleration of the mass relative to the masses
, ', '' ... at distances , ',
'' respectively, is zero or
The latter expression is equivalent to the former statement, as soon as
a sufficient number and sufficiently great and extensive masses are
taken into consideration...." This formulation cannot satisfy us. For,
in addition to a certain requisite accuracy, the character of a
"contact" law is lacking, so that its promotion to the rank of a
fundamental law (in place of the law of inertia) is quite out of the
question.
The inner ground of these difficulties is without doubt to be found in an insufficient connection between fundamental principles and observation. As a matter of actual fact, we only observe the motions of bodies relatively to one another, and these are never absolutely rectilinear nor uniform. Pure inertial motion is thus a conception deduced by abstraction from a mental experiment—a mere fiction.
However necessary and fruitful a mental experiment may often be, there is the ever-present danger that an abstraction which has been carried unduly far loses sight of the physical contents of its underlying notions. And so it is in this case. If there is no meaning for our understanding in talking of the "motion of a body" in space, as long as there is only this one body present, is there any meaning in granting the body attributes such as inertial mass, which arise only from our observation of several bodies, moving relatively to one another? If not, then we cannot attach to the conception "inertial mass of a body," an absolute significance, that is, a meaning which is independent of all other physical conditions, as has hitherto been done. Such doubts received fresh strength when the special theory of relativity endowed every form of energy with inertia (vide Note 17).
The results of the special theory of relativity entirely unhinged our view of the inertia of matter, for they robbed the theorem concerning the equality of inertial and gravitational mass of its strict validity. A body was now to have an inertial mass varying with its contained internal energy, without its gravitational mass being altered. But the mass of a body had always been ascertained from its weight, without any inconsistencies manifesting themselves (vide Note 18).
A difficulty of such a fundamental character could come to light only owing to the theorem of the equality of inertial and gravitational mass not being sufficiently interwoven with the underlying principles of mechanics, and because, in the foundations of Newtonian mechanics, the same importance had not been accorded to gravitational phenomena as to inertial phenomena, which, judged from the standpoint of experience, must be claimed. Gravitation, as a force acting at a distance, is, on the contrary, introduced only as a special force for a limited range of phenomena: and the surprising fact of the equality of inertial and gravitational mass, valid at all times and in all places, receives no further attention. One must, therefore, substitute for the law of inertia a fundamental law which comprises inertial and gravitational phenomena. This can be brought about by a consistent application of the principle of the relativity of all motions, as Einstein has recognized. This is, therefore, the circumstance chosen by Einstein as a nucleus about which to weave his developments.
The theorem of the equality of inertial and gravitational mass, which reflects the intimate connection between inertial and gravitational phenomena, may be illuminated from another point of view, and thereby discloses its close relationship (vide page 55) to the general principle of relativity.
However much the notion of "absolute space" repelled Newton, he nevertheless believed he had a strong argument, in support of the existence of absolute space, in the phenomenon of centrifugal forces. When a body rotates, centrifugal forces make their appearance. Their presence in a body alone, without any other visible body being present, enables one to demonstrate the fact that it is in rotation. Even if the earth were perpetually enveloped in an opaque sheet of cloud, one would be able to establish its daily rotation about its axis by means of Foucault's pendulum-experiment. This peculiarity of rotations led Newton to conclude that absolute motions exist. From the purely kinematical point of view, however, the rotation of the earth is not to be distinguished in any way from a translation; in this case, too, we observe only the relative motions of bodies, and might just as well imagine that all bodies in the universe revolve around the earth. E. Mach has, in fact, affirmed that both events are equivalent, not only kinematically, but also dynamically: it must, however, then be assumed that the centrifugal forces, which are observed at the surface of the earth, would also arise, equal in quantity and similar in their manifestations, from the gravitational effect of all bodies in their entirety, if these revolved around the supposedly fixed earth (vide Note 19).
The justification for this view, which in the first place arises out of the kinematical standpoint, is, in the main, to be sought in the fact, derived from experience, that inertial and gravitational mass are equal. According to the conceptions, which have hitherto prevailed, the centrifugal forces axe called into play by the inertia of the rotating body (or rather by the inertia of the separate points of mass, which continually strive to follow the bent of their inertia, and, therefore, express the tendency to fly off at a tangent to the path in which they are constrained to move). The field of centrifugal forces is, therefore, an inertial field (vide Note 20). The possibility of regarding it equally well as a gravitational field—and we do that, as soon as we also assert the relativity of rotations dynamically: for we must then assume that the whole of the masses describing paths about the (supposed) fixed body induce the so-called centrifugal forces by means of their gravitational action—is founded on the equality of inertial and gravitational mass, a fact which Eötvös has established with extraordinary precision by making use of the centrifugal forces of the rotating earth (vide Note 21). From these considerations one realizes how a general principle of the relativity of all motions simultaneously implies a theory of gravitational fields.
From these remarks one inevitably gains the impression that a construction of mechanics upon an entirely new basis is an absolute necessity. There is no hope of a satisfactory formulation of the law of inertia without taking into account the relativity of all motions, and hence just as little hope of banishing the unwelcome conception of absolute motion out of mechanics; moreover, the discovery of the inertia of energy has taught us facts which refuse to fit into the existing system, and necessitate a revision of the foundations of mechanics. The condition which must be imposed at the very outset (cf. page 20) is: Elimination of all actions which are supposed to take place "at a distance" and of all quantities which are not capable of direct observation, out of the fundamental laws; i.e. the setting-up of a differential equation which comprises the motion of a body under the influence of both inertia and gravity and symbolically expresses the relativity of all motions. This condition is completely satisfied by Einstein's theory of gravitation and the general theory of relativity. The sacrifice, which we have to make in accepting them, is to renounce the hypothesis, which is certainly deeply rooted, that all physical events take place in space whose measure-relations (geometry) are given to us a priori, independently of all physical knowledge. As we shall see in the following section, the general theory of relativity leads us, rather, to the view that we may regard the metrical conditions in the neighbourhood of bodies as being conditioned by their gravitation. In this way the geometry of the measuring physicist becomes intimately welded with the other branches of physics.
If we compress into a short statement what we have so far deduced out of the fundamental postulates formulated at the beginning, we may say: The postulate of general relativity demands that the fundamental laws be independent of the particular choice of the co-ordinates of reference. But the Euclidean line-element does not preserve its form after any arbitrary change of the co-ordinates of reference. We have, therefore, to substitute in its place the general line-element: Whereas, then, the postulate of continuity (cf. page 20) seemed to render it only advisable not to introduce the narrowing assumptions of the Euclidean determination of measure, the principle of general relativity no longer leaves us any choice.
The reason for so emphasizing the latter principle—as, indeed, also the postulate that only observable quantities are to occur in physical laws—is not to be sought in any requirement of a merely formal nature, but rather in an endeavour to invest the principle of causality with the authority of a law which holds good in the world of actual physical experience. The postulate of the relativity of all motions receives its true value only in the light of the theory of knowledge (Note 22). One must, above all, avoid introducing into physical laws, side by side with observable quantities, hypotheses which are purely fictitious in character, as e.g. the space of Newton's mechanics. Otherwise the principle of causality would not give us any real information about causes and effects, i.e. the causal relations of the contents of direct experience; which is, presumably, the aim of every physical description of natural phenomena.