The evolution of scientific thought from Newton to Einstein

by A. D'Abro

[78] There are cases in which no determinate value for the potential can be found.

[79] Note how the theory of relativity is establishing the fusion of the mathematical (i.e., the

-distribution) with the physical (i.e., the forces of inertia).

[80] The statement that all bodies fall with the same motion in vacuo is correct, but, unless properly understood, is apt to lead to erroneous conclusions.

What the statement asserts is that any body, regardless of its constitution or mass, will fall in exactly the same way through a given gravitational field. If, however, the falling body causes a modification in the distribution of the field, then it is obvious that various falling bodies will no longer be situated in the same gravitational field; and there is no reason to assert that they will all fall in exactly the same way. For instance, if a relatively small mass—say, a billiard ball—then a relatively large one—say, the moon—were to be released in succession from some very distant point and allowed to fall towards the earth, it is undoubtedly correct to state that the duration of fall would be greater for the billiard ball than for the moon. It is easy to understand why this discrepancy would arise. In either case, owing to the mutual gravitational action, the earth would also be moving towards the falling body (moon or billiard ball), so that the point of collision would be some intermediate point, namely, the centre of gravity of the system earth-moon or earth-billiard ball. But in the case of the billiard ball, owing to its relatively insignificant mass, this centre of gravity, or point of collision, would be practically identical with the earth’s centre. This implies that the earth would scarcely move at all towards the billiard ball, whereas it would move an appreciable distance towards the moon. The motion of the earth would thus shorten considerably the distance through which the moon would have to fall, whereas the billiard ball would have to fall through the entire distance.

If we wish so to modify the conditions of the problem as to re-establish the perfect identity in the rate of fall of the moon and billiard ball, we must so arrange matters that the earth is unable to fall towards the body it is attracting. If, for example, it were possible to nail the earth to the Galilean frame in which earth, moon and billiard ball were originally at rest, and if this Galilean frame could be made to remain Galilean, i.e., unaccelerated, then the previous experiment attempted first with the moon, then with the billiard ball (the moon being removed entirely), would reveal exactly the same rate of fall for the two bodies. For now, indeed, the modifications in the distance covered, and in the nature of the field brought about by the displacement of the earth, would be non-existent.

A further case which presents a theoretical interest in Einstein’s discussions is afforded by what is known as a uniform field, that is to say, a field in which the gravitational force is the same in intensity and direction throughout space. A field of this sort would be generated by an infinitely extended sheet of matter of uniform density. Owing to the infinite mass a sheet of this sort would possess, its acceleration towards the falling body would be nil; all bodies would then fall with exactly the same constant acceleration towards the sheet. The reason uniform fields present a theoretical interest is because the field of force generated in an enclosure moving with constant acceleration is precisely of this type. When, therefore, Einstein identifies the field of force enduring in an accelerated enclosure with a gravitational field, we must remember that the distribution of matter which would be necessary to produce the same field is that of an infinitely extended sheet. Only to a first approximation can a finite mass of matter, like the earth, be deemed to generate a field of this kind.

[81] Subject to certain niceties which will be mentioned presently.

[82] To obviate any confusion with electromagnetic forces, we are considering only forces which act on uncharged bodies.