. The result is quite general and applies to all processes. Thus, the duration of the revolution of a top will be possessed of varying values according to the relative motion of the observer. Arguments similar to those we elaborated in the case of space go to show that time is no longer unique. A multiplicity of different time directions, variously tilted, must coexist, each one referring to some particular frame of reference, hence to some particular observer. As a result, the simultaneity of two events occurring at different points of space can no longer be absolute, since simultaneity depends essentially on the time which we select, and therefore on our frame of reference.
Just as each frame had its own space, so also would it now have its own time and its own definition of simultaneity. We may summarise these various discoveries by stating that there can exist no universal definition of practical congruence either for space or for time; there can exist, therefore, no geometry for a universal space or for a universal time.
The philosophical consequence of these new points of view is to deprive universal space and time of the objective significance with which they were formerly credited. Henceforth, in the words of Minkowski, “space and time, by themselves, sink to the position of mere shadows.”
Thus, consider a yardstick. If this same yardstick is measured by various observers passing it with different velocities, it will be possessed of different lengths. The space this rod occupies can then be credited with no particular magnitude, since this magnitude itself depends as much on the point of view of the observer as on the characteristics of the rod. This is what is meant by the relativity of distance. We must not confuse this discovery of Einstein’s with the fundamental relativity of all distance in conceptual space. For while classical science recognised that there was no such thing as absolute distance in conceptual space, yet it believed that in the real space of the physicist the distance between two given points in a frame of reference could be defined unambiguously for all observers, by means of rigid rods. It is this last opinion which is shattered by Einstein’s discoveries.
It is the same for duration when we remember that the duration of any phenomenon will be possessed of varying values according to the relative motion of the observer; and it is this fact which is expressed by the statement that time or duration is relative.
Thus, neither space nor time by itself can exist in the real phenomenal world which the physicist explores. Space and time appear as mere modes of perception, mere relations, varying and changing according to the conditions of relative motion existing between the observer and the observed. With the disappearance of the objectivity of space and time regarded as absolute and universal forms of our perception, the objectivity of the entire physical world seemed to sink into a mist. If, then, any kind of objectivity was to be retained, it was necessary to effect a synthesis of all the individual points of view of all the different observers, to mould into one sole representation the multiplicity of individual spaces and the multiplicity of individual durations.
Could this result be accomplished, the relativity of practical congruence for both space and time would give way to some universal definition of practical congruence holding for all observers regardless of their relative motion, and connoting, therefore, the existence of some absolute continuum no longer a mere shadow. In place of the individual point of view varying with the relative motions of the observers, we should obtain an impersonal and hence common objective understanding of nature.
The achievement of this supreme synthesis of the points of view of all observers was accomplished by Minkowski in 1908. He succeeded in obtaining an invariant definition of congruence by combining any given observer’s definitions of practical congruence for space and for time, and by showing that this combination possessed an invariant value holding equally for all other observers. Of course, the type of congruence obtained was one neither of space nor of time. It was a combination of both. But its existence showed us that, transcending space and time, there existed an impersonal and fundamental four-dimensional metrical continuum which Minkowski called space-time. The definition of congruence obtained by him for this mysterious continuum proved it to be Euclidean (more properly, semi-Euclidean).
The elusiveness of space and time, or rather the ambiguity of these concepts, depending as they do on the observer who measures and senses them, is replaced by the common objectivity of this fundamental continuum, which is the same for all and transcends the particular conditions of motion of the observer. This continuum, though of itself neither space nor time, yet pertains to both in that the observer is able to carve it up into that particular space and that particular time which are characteristic of the frame of reference in which he is stationed, and which constitute the space and time in which he lives and experiments.
From now on, the real extension of the world has a fourfold order, and the geometry of relativity is necessarily four-dimensional. What we call the space of our Galilean frame at an instant, is but a cross-section of this four-dimensional space-time world; and what we call the time of our frame lies along a perpendicular to this section. As we pass from one Galilean frame to another, we change our space and we change our time. Our new instantaneous space, though still a three-dimensional cross-section of the four-dimensional space-time world, is tilted with respect to our former cross-section; and the same tilting ensues for the perpendicular direction called time. In fact, the situation presents a striking analogy with the various directions of the verticals to the earth’s surface. They, also, manifest different directions because the various two-dimensional portions of the earth’s surface are variously tilted in three-dimensional space. The great change in our understanding of the spatio-temporal background, the great difference between space-time and the separate space and time of classical science, resides, therefore, in this tilting process which accompanies a change of relative motion.