Einstein's Theories of Relativity and Gravitation / A selection of material from the essays submitted in the competition for the Eugene Higgins prize of $5,000

by J. Malcolm Bird

The Genesis of the Theory

An experiment by Michelson and Morley (1887), on which the whole theory is based, made it appear that if a man measures the velocity at which light passes him he will get the same result whether he is stationary, rushing to meet the light, or moving in the same direction as the light. The solution was provided by Einstein in 1905. He suggested that since we know the results of these determinations ought not to agree, something must have happened to the clocks and measuring-rods used in measuring the velocity so that the standards of length and time were not the same in the three cases, the alterations being exactly such as to make the velocity of light constant. This solution is universally accepted as true and is the fundamental postulate. Thus the length of a stick and the rate at which time passes will change as the velocity of the person observing these things changes. If a man measured the length of an aeroplane going past him at 161,000 miles per second it would measure only half the length observed when stationary. If the aeroplane were going with the velocity of light, its length would vanish though its breadth and height would be unaltered. Similarly, if of two twin brothers one were continually moving with reference to the other their ages would gradually diverge, for time would go at different rates for the two. If one moved with the velocity of light, time would stand still for him while for the other it would go on as usual. To get actually younger it would be necessary to move quicker than light which is believed to be impossible. The velocity of light is assumed to be the greatest velocity occurring in nature.

Evidently then if the distance in space and the interval in time separating two given events, such as the firing of a gun and the bursting of the shell, are measured by two observers in uniform relative motion, their estimates will not agree. Consider now the simple problem of measuring the distance between two points on an ordinary drawing-board. If we draw two perpendicular axes, we can define this distance by specifying the lengths of the projections on the two axes of the line joining the points. If we choose two different axes the projections will not be the same but will define the same length. Similarly, in a Euclidean four-space the distance between two points will be defined by the projections on the four axes, but if these axes be rotated slightly, the projections will be different, but will define the same length. Now, returning to the two observers just mentioned, it was noticed by Minkowski in 1908 that if the space measurements between the two events are split into the usual three components, and if the time measurements are multiplied by

, the difference between the two sets of measurements is exactly the same as would have occurred had these two events been points in a Euclidean fourspace, and two different observations made of their distance apart using two sets of axes inclined to each other. The velocity of light is made equal to 1 in this calculation by a suitable choice of units. This discovery threw a vivid light on the problem of space-time, showing that it is probably a true four-space of one negative dimension, a simple derivative of the much-discussed and now familiar Euclidean four-space.

Although this discovery gave a tremendous impetus to the progress of the theory, it is probable that it holds a deeper significance not yet revealed. It is probably a statement of the “stuff” of which the four-space is made, and perhaps also of how it is made; but the problem remains unsolved.

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.