would automatically yield us the invariant velocity. In particular, we might measure the ratio of an electric charge when computed in terms of electrostatic, then of electromagnetic units; this again would yield us the value of Maxwell’s constant
, hence of the critical invariant velocity. We might also appeal to the necessary consequences of the existence of a critical velocity, for instance, to the peculiar Lobatchewskian law which it would entail for the additions of velocities. We should then check these consequences by experiment, discover whether they were justified by facts, and deduce therefrom the existence and value of the invariant velocity.
Fizeau’s experiment permits this verification and proves that the composition of velocities is Lobatchewskian, not Euclidean, and corresponds to the existence of a critical invariant velocity of 186,000 miles per second. It is true that in Fizeau’s experiment we are again dealing with the propagation of light, though no longer in vacuo, where its speed is invariant, but through water. It is precisely because its speed through water is reduced that the velocity of the water will affect the velocity of the light, thereby permitting us to test out the relativistic law of the composition of velocities. But once again the reason why light propagations enter into the majority of these tests is due solely to the fact that optical experiments permit of greater accuracy. Apart from this, there is no need to limit ourselves to experiments dealing with the propagation of light, whether in vacuo or through material bodies such as water. Any speed, be it the speed of sound or of a bullet shot from a passing train, should enable us to detect the same law of Lobatchewskian composition and thus allow us to demonstrate the existence of space-time.
At all events, one point should be clear by now. Whatever method we follow, we can never get away from performing physical measurements of one kind or another. It is therefore meaningless to speak of defining the invariant critical velocity without reference to a measurement of the speed of light in vacuo, or more generally, without reference to physical measurements bearing on the phenomena which the existence of a finite invariant speed in the universe would necessarily entail. From this it follows that without physical measurements of one kind or another we have no means of establishing whether the world is one of separate space and time or one of space-time.
In theory even crude observations should have led us to a decision on this score, for crude observations differ from those of the physicist only in the matter of precision. Theoretically, therefore, prior to Einstein we should have known of the incorrectness of the Euclidean composition of velocities and hence known of space-time. The reason for our failure to make the momentous discovery is easily understood when we realise that owing to the very great magnitude of the invariant finite velocity
, the Einsteinian rule can scarcely be differentiated from the classical rule unless very great velocities are involved. From this it is seen that the reason why the effects anticipated by Einstein’s theory are so difficult to detect is because of the very high value of the invariant velocity
. Had this finite invariant velocity happened to be small, say a mile a second, the classical conception of a world of separate space and time would have been abandoned years ago; for the curious effects predicted by Einstein would have become perceptible without effort. Again, were we to conceive of this invariant velocity as becoming greater and greater, the distinguishing features of the Einsteinian universe would gradually fade away, until finally, if the invariant velocity became infinite, space-time would give way to the separate space and time of classical science, and all the peculiarities of Einstein’s universe would vanish.