. As before, though
appears longer than
, it is, in reality, shorter, as can be understood by referring to [Fig. XIX]. Incidentally, we see how the absolute nature of acceleration causes the traveller’s world-line to bend; and it is this absolute bend in the world-line which differentiates the life-histories of the two twins and which is responsible for an absolute, non-reciprocal difference in their respective aging.
Let us also state that just as was the case with the classical graph which represented geometrically the classical Galilean transformations, so now the Minkowski graph merely translates the Lorentz-Einstein transformations.
And here a matter of some importance must be noted. We remember that before proceeding to draw our graph, we were compelled to settle on a co-ordination of units of measurement for space and time. We then chose the same length on our sheet of paper to represent a duration of one second and a distance of 186,000 miles. As a result of this choice the world-lines of light rays became diagonals. But suppose we had selected other units. Then, obviously, the slant of the world-lines, hence the angle of the light-cone, would have been modified and the entire appearance of the graph changed. Inasmuch as our choice of units is entirely arbitrary, we might be led to believe that the graph could not depict reality. But this opinion would be unfounded. While it is true that owing to the arbitrariness of our units, the graph cannot aspire to represent absolute shape, yet it does express certain definite relationships which a change of units could not disturb. In fact we might conceive the graph to be distorted by stretching, but still the relationships would endure; and relationships are all that science (or, we might even say, the human mind) can ever aspire to approach.
On the other hand, this question of units allows us to give a graphical solution of a point which is of great philosophical interest. Here we are living in a world which, theoretically at least, is vastly different from the world of separate space and time, and yet it is only thanks to ultra-refined experiment and to the genius of Einstein and Minkowski that we have finally realised it to be a four-dimensional continuum of events. How is it that ordinary perception is so blind to facts?
In order to understand this point, we must mention that though our choice of units for co-ordinating space and time measurements is arbitrary, since there is no rational connection between the magnitude of a distance in space and that of a duration in time, yet our daily activities suggest a common standard of comparison. The fact is that the distances which we ourselves and other material bodies cover in one second over the earth’s surface are always comprised within certain narrow limits. This leads us to couple one second and one yard, rather than one second and 186,000 miles.