Possibly further illustrations of space-time may be helpful. Thus, if we consider the three-dimensional extension occupied by the atmosphere and note the variations in temperature that occur from place to place, we are justified in speaking of a four-dimensional continuum space-temperature But it is obvious that we are at liberty to divorce these two entities and study them separately, obtaining thereby two separate continua, one of space and one of temperature. In classical science, it was much the same with space and time. We could consider them jointly as constituting a four-dimensional continuum of events but then again, owing to the absolute nature of time and distance, we could divorce them. When this was done, both three-dimensional space and uni-dimensional time were separated into distinct self-supporting continua.
With the space-time of relativity, a separation of this sort is impossible. We cannot withdraw the time dimension, for it is no longer unique. If we cancel it so far as some definite observer is concerned, leaving him with an instantaneous space, we shall have mutilated the instantaneous space, without having cancelled time, in the estimation of another ob server in relative motion. We see, then, that whereas in classical science the adjunction of time to space was artificial, in that time and space could always be separated like oil and vinegar, such is no longer the case in relativity. Space and time are now blended together and can be claimed to constitute one solid whole. In a very crude way, we may say that space-time is more akin to an opal emitting various colours in various directions. Just as with space-time it was impossible to annihilate time or space for all observers, so here, with the opal, it would be impossible to cancel all the points that reflected green light and only green light since if we pursued this course from the standpoint of one observer we should have failed to achieve our object from the standpoint of another.
Let us now consider the characteristics of metrical continua. From a mathematical standpoint, a metrical continuum is one with which it is possible to associate a geometry, hence to obtain an invariant expression for the distance separating two points of the continuum, the value of this expression holding for all observers. In the case of space-time, this would imply that the distance between any two point-events should exhibit the same space-time separation holding for all observers regardless of their motion. Now, in the space and time of classical science, no such invariant space and time distance was possible. A simple example may make this point clear. Suppose that in classical science we were standing on a railroad embankment, and viewing the production of two instantaneous events—say, two light flashes emitted by two spatially separated signals on the line. By taking into consideration the speed with which the light signals had progressed towards us and by measuring our distance from the respective signals, we could determine the spatial distance between the two signals and the respective times at which the flashes had occurred. We should thus obtain the temporal separation and the spatial separation of the two events as referred to the embankment. But suppose now that we had viewed the same two events from a train moving uniformly along the track. In classical science, time being absolute, the temporal separation between the two events would, of course, suffer no modification. The spatial separation as referred to the train would, however, generally be different. We can understand this by supposing that the train passed before the two successive signals as their respective flashes were emitted; in this case, as referred to the train, the two events would have occurred at the same point of the train, and so their spatial separation would be zero. We see, then, that the time separation would be the same in either case, whereas the space separation would have varied. As a result, the space and time separation could not remain invariant. For the separation between the two point-events to manifest itself as invariant, we may surmise that the time separation should also vary in such a way as to offset the variation of the space separation. The absoluteness of time in classical science rendered this impossible, and this is what is meant by stating that space and time were not amalgamated into one continuum in any deep sense.
Now, what Einstein tells us is that had we performed our measurements with extreme accuracy, using our customary chronometers and rigid rods, we should have found that the time separation between the two events had varied slightly; and that as a result this slight variation would have cancelled the corresponding change of the space separation, leaving the space-time distance unmodified.
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