Page:Relativity (1931).djvu/86

 is naturally four-dimensional in the space-time sense. For it is composed of individual events, each of which is described by four numbers, namely, three space co-ordinates $$x$$, $$y$$, $$z$$ and a time co-ordinate, the time-value $$t$$. The “world” is in this sense also a continuum; for to every event there are as many ‘‘neighbouring” events (realised or at least thinkable) as we care to choose, the co-ordinates $$x_1$$, $$y_1$$, $$z_1$$, $$t_1$$ of which differ by an indefinitely small amount from those of the event $$x$$, $$y$$, $$z$$, $$t$$ originally considered. That we have not been accustomed to regard the world in this sense as a four-dimensional continuum is due to the fact that in physics, before the advent of the theory of relativity, time played a different and more independent rôle, as compared with the space co-ordinates. It is for this reason that we have been in the habit of treating time as an independent continuum. As a matter of fact, according to classical mechanics, time is absolute, i.e. it is independent of the position and the condition of motion of the system of co-ordinates. We see this expressed in the last equation of the Galileian transformation $$( t' = t )$$.

The four-dimensional mode of consideration of the “world” is natural on the theory of relativity, since according to this theory time is robbed of its independence. This is shown by the fourth equation of the Lorentz transformation: