Thus all the problems we have discussed could be solved either geometrically or analytically.
It is a matter of common knowledge that this space and time graph we have discussed was rarely mentioned in classical science; and such appellations as point-events and world-lines were unknown prior to Minkowski’s discoveries. It may seem strange, therefore, that the theory of relativity should have made so extensive a use of a graphical method of representation. The reason is that in classical science the graph reduced to a mere geometrical superposition of space and time which was convenient in certain cases, but which had no profound significance. By reason of the separateness of space and time exemplified by the absoluteness of the space and time axes, there existed no amalgamation, no unity between space and time. They did not form one four-dimensional continuum of events in any deep sense. Let us endeavour to understand the meaning of these statements.
In classical science, space was regarded as a three-dimensional continuum of points, because it was possible to localise the position of a point in our frame by referring to its three co-ordinates. Suppose, then, that we wished to measure the distance between two points in space. All we should have to do would be to stretch a tape between the two points, and the length of the tape would define the distance between them.
And now let us consider the space and time of classical science. Here, again, we might claim that space and time constituted a four-dimensional continuum of events; for we could always localise the occurrence of an instantaneous event by measuring three spatial co-ordinates in our frame of reference, and by computing the instant at which the event occurred. But suppose we wished to measure the distance between two events occurring, say, one in New York on Monday, and the other in Washington on Tuesday. Obviously the problem would be meaningless. We might say that the distance in space between the two events was so many miles, and their distance in time so many hours; but we should be unable to measure the distance in space-time itself directly, whereas, with the aid of the tape, in our previous example, we were able to measure immediately the spatial distance between the two points.
We see, then, that whereas both classical space and classical space-cum-time constituted a three-dimensional and a four-dimensional continuum, respectively, yet there was a vast difference between these two continua. The first one was a metrical continuum, whereas classical space-cum-time had no four-dimensional metrics.
We may present these arguments in a slightly different way, as follows: We say that a necessary condition for a continuum of points to form a metrical continuum possessing a definite geometry is that a definite unambiguous expression, invariant in magnitude, be found for the distance between any two points. Now this condition is certainly not realised in our classical space and time graph.
Fig. XV
Consider, as before, the embankment, an observer at rest at
