It thus becomes plain that our two observers are merely looking at the same thing from different viewpoints. Each has just as much right as the other to regard himself as being at rest in ordinary space (this is the postulate of the relativity of uniform motion) and to regard his time direction as a straight line in the four-space. The difference is merely that the two time axes are inclined to each other. If, however, one were moving with an acceleration with reference to the other his path in the four-space will appear curved to the other, though he himself, since he regards it as his time axis, will still assume it to be straight. If there is a body moving in what one observer sees to be a straight line, the other will, of course, in general see it as curved, and following the usual custom, since this body, without apparent reason, deviates from the straight path, will say there must be some force acting on it. Thus the curvature of his time axis, due to his accelerated motion, makes it appear that there is round him a field of force, which causes freely moving bodies to deviate from the straight path. Now if space-time is itself inherently curved it is not generally possible for any line in it to be straight any more than it is possible for any line on the surface of a sphere to be straight. Hence, all axes must be curved, and all observers, whatever their states of motion, must experience fields of force which are of the same nature as those due to motion only. The extra force experienced when a lift begins to rise is an example of force due to pure motion: gravitation is the similar force due to an inherent curvature of the four-space, and it was the postulate that these forces were similar that made possible Einstein’s solution of the general problem of gravitation.
The Time Diagram
The correlation of time with its geometrical analogue is of absorbing interest. Representing velocity by the common method of plotting a curve showing positions at various times and marking distances horizontally and times vertically, the velocity of light being 1, MM′ and NN′ will both represent this velocity. Since this is assumed to be the greatest velocity occurring in nature, all other possible velocities
are represented by lines falling within the upper and lower V’s. Now this diagram correctly represents two dimensions of Minkowski’s Euclidean four-space so, transmuting to real but flat four-space by multiplying times by
, it is seen that there is a region outside which no effect can be propagated from O since that would involve the existence of a velocity greater than that of light. This region represents the future of O. Similarly, O can only be affected by events within the region derived from the downward-opening V, which therefore represents the past of O. The region between the two represents events which may be either simultaneous with O or not, according to the velocity of the observer at O. Thus in this theory an event dictated by free-will, could affect points in its “future” region, but not in any other, which agrees with experience and shows that the theory is not essentially “determinist.” If “free-will” is really free, the future is not yet determined, and the fourspace must be in some way formed by the will as time progresses.
The trains of thought inspired by Einstein’s postulates have already carried us to a pinnacle of knowledge unprecedented in the history of man. On every hand, as we look out upon the universe from our new and lofty standpoint, unexpected and enthralling vistas open up before us, and we find ourselves confronting nature with an insight such as no man has ever before dared aspire to.
It is completely unthinkable that this theory can ever be swept aside. Apart from experimental verifications which, in point of fact, lend it the strongest support, no one could work through the theory without feeling that here, in truth, the inner workings of the universe were laid bare before him. The harmony with nature is far too complete for any doubt to arise of its truth.