The evolution of scientific thought from Newton to Einstein

by A. D'Abro

However, it is well to caution the beginner against a misunderstanding which might cause him some trouble. In classical science the word “gravitational” was always associated with the attraction caused by matter. In Einstein’s theory, when we identify an inertial field with a gravitational one and call both these species of fields gravitational, it is not meant to imply that an exact replica of the inertial field could be reproduced by disposing matter in a suitable way with respect to our frame of reference. Inversely, a field of force produced by matter cannot be duplicated in every detail by communicating a suitable accelerated motion to our frame of reference in free space far from matter. In spite of the complete identification of forces of gravitation and forces of inertia, there still exists a difference in the spatial distribution of the field of force, according to whether it is produced by the proximity of matter or by the acceleration of our frame. In consequence, although all fields of force may be called gravitational fields, regardless of whether they be generated by matter or by acceleration, we must remember that the actual lay of these fields through space will vary with their origin.

A few precise illustrations will make this point clearer. Consider a hollow chest pulled by some unseen hand through interstellar space far from matter. If we suppose that the chest is rising vertically with constant acceleration, we know that as a result of this constant acceleration a uniform field of inertial force will be present in the interior of the chest. The postulate of equivalence consists in asserting that the observer in the interior of the chest might with equal justification consider the chest to be unaccelerated or at rest in space, while the field of force he perceives would be assimilated to a field of gravitation. All that it is meant here to imply is that the physical nature of fields of gravity and of inertia is one and the same. It is not meant to imply that any actual distribution of matter could ever produce a perfectly uniform field of force in the chest’s interior such as existed under the influence of constant acceleration. We know, indeed, that a distribution of matter under the chest would produce a somewhat similar field of force, but the field would be non-uniform, the magnitude of the force being smaller at the top than at the base of the chest. Thus, although the physical nature of both types of fields would be the same, the precise distribution of these fields through space would be different.

These, at least, are the conclusions by which we must abide at the present stage of the theory. We shall see that when we come to consider the universe as a whole, there may be grounds for modifying our opinions. But at the present stage of the discussion we must admit that fields of force produced by matter can never be distributed in exactly the same way as fields produced by acceleration, and vice versa.

Another example is afforded by the field of force generated on a rotating disk. This field of force can be split up for purposes of analysis into two separate fields: a centrifugal field constituted by forces directed away from the centre of the disk, and a Coriolis field pulling bodies sideways as they approach or move away from the centre. Now, no distribution of matter around the disk can be conceived of at this stage which would produce a field of gravitation on the disk distributed in exactly the same way as this field of inertia. Nevertheless, the inertial field on the disk can be called a field of gravitation for the reasons previously set forth in this chapter.

Again, just as matter cannot produce a field of force distributed in exactly the same way as a field generated by acceleration, so also is it impossible to produce by the sole virtue of acceleration a field of force distributed in exactly the same way as the gravitational field around matter. From this it follows that it is impossible, by merely selecting some suitably accelerated frame, to cancel in its entirety a field of force produced by matter. This statement may appear to be in conflict with the example of the falling elevator, in which it was explained that owing to the motion of the elevator the field of force produced by the earth vanished in its interior. However, there is no conflict between the two statements, if we are careful to express ourselves with precision.

In the falling elevator no field of force is experienced in the interior of the elevator, provided the extension of the elevator is small in comparison with the size of the earth. If the elevator were very large, if, for instance, it were a gigantic box completely surrounding the earth, no conceivable motion of the box could ever cause the gravitational field of the earth to vanish everywhere in the box’s interior. Suppose, for instance, that the roof of the box were accelerated towards the North Pole, its floor would then be moving with accelerated motion away from the South Pole. Hence, whereas an observer attached to the roof of the box would perceive no force of gravitation, an observer attached to the floor would experience a force of doubled intensity. In other words, it would be impossible to banish the force of gravitation in all parts of the box. Only when the box or elevator is small in comparison with the earth, so that within its confines the field produced by our planet is practically uniform, is it possible to cancel the earth’s field in the whole interior of the box. Even then, however, this cancellation would be rigorous only at a point, and would decrease in thoroughness as we moved away from this point.

It follows that a falling elevator cannot be assimilated in all rigour to a Galilean frame, since in a Galilean frame, wherever we might be stationed, no forces would be experienced. Nevertheless, as in practice, the frames we consider are not supposed to extend for great distances in all directions, conditions enduring in a falling elevator would be identical to a very high order of precision with those existing in a Galilean frame. For this reason, rigid frames of small dimensions falling freely in a gravitational field are termed semi-Galilean.

If we summarise the conclusions reached in this chapter, we may say that the postulate of equivalence has allowed us to identify forces of inertia with forces of gravitation. But this identification applies solely to the nature of the forces, not to their spatial distribution. And so it is not correct to say that it would be impossible for us to ascertain whether the field of force experienced in our enclosure was due to the acceleration of the enclosure or to the proximity of gravitating masses. Even without peering out and discovering whether large masses were present, we could always, at least in theory, by a mere exploration of the field distribution, ascertain the true conditions.

We see, then, that whereas velocity was relative in that it was quite impossible for us to ascertain the absolute velocity of our enclosure, acceleration still remains absolute in spite of the postulate, since (theoretically at least) it can always be detected and its effects separated from those due to matter. In fact, as we shall see later, the complete relativity of all motion can be established only if the universe is finite. Under the circumstances, we must be careful not to overestimate the philosophical significance of the postulate in its bearing on the relativity of motion.

Now it might be thought that owing to this fundamental difference in the spatial distribution of fields of force (inertial and gravitational), the postulate would not be of much use in its physical applications. But this view would be erroneous. From a purely qualitative standpoint the postulate permits us to assert that any phenomenon whose behaviour should be affected by the acceleration of the enclosure, must also be sensitive to the presence of a gravitational field due to matter. Inasmuch as it is often easy to see that the acceleration of our frame of reference must inevitably modify the observed behaviour of a phenomenon, we are able to infer therefrom that the same phenomenon will also be affected by a gravitational field generated by matter.