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. [Pg 201] 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 . Obviously , being a maximum velocity, would have to be greater than or equal to the highest velocity known to physicists, hence would have to be equal to or greater than , velocity of light. Let us therefore suppose that is greater than . As in Einstein’s theory, we should still be faced with a world of four-dimensional space-time, but this world would differ from that of relativity, owing to the discrepancy existing between the invariant velocity and Maxwell’s constant . This would indicate that the equations of electromagnetics would no longer maintain the same form when we changed Galilean frames. As a result, the erstwhile negative experiments in electromagnetics should now yield positive results, and absolute velocity through space would be detected. The theory of relativity would have to be abandoned. [Pg 202] But we may consider still another case. Suppose that the invariant velocity of the world were some imaginary quantity , where stands, as before, for , and for some arbitrary finite number. The square of this imaginary velocity being a negative number, the square of would now assume the form