in all directions, provided this speed be measured by the observer at rest in the frame, and not by some other observer in some other frame, whether Galilean or accelerated.[52] And here we must recall that classical science, believing in the absoluteness of time, assumed that events which occurred “now” anywhere throughout space would continue to occur in the same simultaneous fashion regardless of the motion of the observer. This belief was equivalent to assuming that a propagation moving with infinite speed with respect to one observer would also advance with infinite speed when measured by any other observer. In other words, an infinite speed was an absolute; it constituted the invariant speed of classical science. No such physical propagation was known to science, though it was suspected that gravitation might be of this type. At any rate, classical science accepted the existence of this infinite invariant velocity in a conceptual way. This implied the existence of the Galilean transformations, hence of the ordinary laws for the addition of velocities. Inasmuch as all measurements with rigid rods, and with clocks such as vibrating atoms, appeared to bear out the anticipations of classical science, at least so far as crude observation could detect, the classical stand appeared to be vindicated a posteriori even though the hypothesis of absolute time presented no a priori justification.
Now, in the theory of relativity, Einstein asks us to agree that a certain finite invariant velocity which turns out to be that of light in vacuo must be considered invariant for all Galilean observers.[53] We are thus led, as explained previously, to the Lorentz-Einstein transformations, an immediate consequence of which is that molar matter can never move with a velocity greater than or even equal to that of light. This in itself is sufficient to exclude our taking into consideration observers moving with any such speeds.
But suppose that for argument’s sake we wish to consider, even though it be in a purely conceptual way, a velocity greater than light—in particular, a velocity which would constitute infinite speed for any definite Galilean observer. How would a velocity of this sort manifest itself to some other Galilean observer in motion with respect to the first?—The sole point it will be necessary to stress is that this erstwhile infinite velocity could never be invariant without coming into conflict with the invariance of the velocity of 186,000 miles per second. This becomes obvious when we consider that, according to classical science, if we added any speed—say, the speed of light
—to the infinite speed, we should obtain the infinite speed; whereas, in Einstein’s theory, we should obtain the speed of light. From this it follows that the premises of classical science and of relativity are incompatible. Inasmuch as it is Einstein’s theory that we propose to discuss, we shall abandon the classical belief in the invariance of the infinite velocity.
Henceforth the invariant velocity of the universe is to be
, or 186,000 miles per second. As for an infinite velocity, this will now become a relative, infinite for one observer, finite for another. Now it is obvious at first sight that if our space and time measurements were such as classical science believed them to be, it would be impossible for a ray of light to pass us with the same speed regardless of whether we were rushing towards it or fleeing away from it. A simple mathematical calculation shows us, however, that we can make our results of measurement compatible with the postulate of invariance provided we recognise that our space and time measurements are slightly different from what classical science had assumed. This is purely a mathematical problem and can be solved by mathematical means. It leads us, of course, to the Lorentz-Einstein transformations; and from these transformations it is easy to see that rods in relative motion must be shortened, durations of phenomena extended, and the simultaneity of spatially separated events disrupted.
Einstein’s premises being consistent (as can be proved mathematically), no question can arise as to the consistency of Einstein’s conclusions; premises and conclusions must all stand or fall together. The major problem, then, is to analyse the legitimacy of the theory from the standpoint of its ability to portray the real world of physical phenomena. Some lay critics, such as Bergson, alarmed by the revolutionary nature of Einstein’s discoveries and animated by a desire to retain their classical belief in the absolute nature of simultaneity remaining the same for all observers, have endeavoured to uphold this intermediary view. They have maintained that from a purely mathematical standpoint the systems of measurement of relativity might confer greater simplicity on the purely mathematical treatment of the equations of electrodynamics; but that in no case was it permissible to assume that the anticipations of the theory could ever correspond to what we should actually measure with rigid rods and ordinary clocks. We may dismiss this view summarily. Were it correct, none of the physical anticipations of Einstein’s theory would have been verified by physical experiments.
Furthermore, a theory such as Einstein’s is one of mathematical physics, not one of pure mathematics. It is not merely necessary that the deductions of the theory be mathematically consistent with the premises. They must also lead us to anticipations which will be verified by experiment. If they failed in this respect, they would still portray the workings of a possible rational world; but the theory itself, yielding erroneous physical previsions, could never allow us to foresee and to foretell, and no scientist would ever dream of upholding it. In short, we must recognise that if Einstein’s premises are accepted, we must assume that measurements conducted with natural rods and natural clocks would yield the previsions of the theory. We must also remember that by natural clocks we do not mean clocks arbitrarily adjusted. We mean clocks such as vibrating atoms, or, more generally, all periodic physical processes evolving in a state of isolation. Likewise, by natural rods we mean our ordinary material rods maintained in the same conditions of temperature and pressure.