We can also understand how it comes that the discovery of space-time permits us to account for the duality in the nature of motion, relative when translationary and uniform, absolute when accelerated or rotationary. As long as we were endeavouring to account for this duality by attributing an appropriate structure to three-dimensional space, we were met by a peculiar difficulty: If we assumed space to have a flat structure, that is, one of planes and straight lines, it would be possible to understand why a free body, when set in motion with any definite velocity, should be guided along one of these straight lines or grooves in space. It would be following its natural course and there would be complete symmetry all around it; everything would run smoothly and no forces would arise. Then, if we attempted to tear the body away from its straight course, compelling it to follow a curve, we should be forcing it to violate the laws of the space-structure, tearing it away from the space-groove which it would normally follow. It was not impossible to assume that the groove would react and that antagonistic forces would arise. Thus the fact that curved or rotationary motion was accompanied by forces of inertia, hence manifested itself as absolute, did not constitute a difficulty of an insuperable order. But a real difficulty arose when we considered the other species of accelerated motion, i.e., variable motion along a straight line. In this case, the paths of the bodies being straight (hence in harmony with the flat structure of space), no violation of the laws of this spatial structure was evidenced. And yet forces of inertia were again present. It was difficult to interpret their appearance in terms of the space-structure.
When, however, we substitute four-dimensional space-time for separate space and time, the difficulty vanishes. For now we notice that all accelerated motions, regardless of whether they be rectilinear or curvilinear, are represented by curved world-lines in space-time. All these accelerated motions violate, therefore, the flat space-time structure; and for this reason forces of inertia will always be generated by them. On the other hand, when we consider Galilean or uniform or translational motions, we see that, as in the case of three-dimensional space, they will be represented by straight lines. These motions will then stand in perfect harmony with the flat space-time structure, and no forces of inertia will arise.
In this way, thanks to space-time, one of the outstanding difficulties which confronted classical science, namely, the dual nature of space and of motion, is accounted for.
CHAPTER XIX
VARIOUS POSSIBLE WORLDS
WE have seen that the great distinction between Einstein’s theory and classical science arises from the value to be attributed to the invariant velocity of the world. In the belief of classical science, this velocity was infinite, and we were thus led to a world of separate space and time. According to relativity, the value of the invariant velocity is finite and is given by
, the constant which enters into Maxwell’s equations; this constant being illustrated physically by the velocity of light in vacuo.
From a purely mathematical standpoint, however, if we disregard the relations of reality, we may consider other purely formal possibilities. Suppose that the invariant velocity, though finite, were some number
differing from Maxwell’s constant