Page:Eddington A. Space Time and Gravitation. 1920.djvu/87

IV] of tracks from $$P_1$$ to $$P_2$$ of zero interval-length; there is just one which has maximum length. This is because of the peculiar geometry which the minus sign of $$(t_2 - t_1)^2$$ introduces. For instance, it will be seen from equation (1), p. 53, that when that is to say when the resultant distance travelled in space is equal to the distance travelled in time, then $$s$$ is zero. This happens when the velocity is unity—the velocity of light. To get from $$P_1$$ to $$P_2$$ by a path of no interval-length, we must simply keep on travelling with the velocity of light, cruising round if necessary, until the moment comes to turn up at $$P_2$$. On the other hand there is evidently an upper limit to the interval-length of the track, because each portion of $$s$$ is always less than the corresponding portion of $$(t_2 - t_1)$$, and $$s$$ can never exceed $$t_2 - t_1$$.

There is a physical interpretation of interval-length along the path of a particle which helps to give a more tangible idea of its meaning. It is the time as perceived by an observer, or measured by a clock, carried on the particle. This is called the proper-time; and, of course, it will not in general agree with the time-reckoning of the independent onlooker who is supposed to be watching the whole proceedings. To prove this, we notice from equation (1) that if $$x_2 = x_1$$, $$y_2 = y_1$$, and $$z_2 = z_1$$, then $$s = t_2 - t_1$$. The condition $$x_2 = x_1$$, etc. means that the particle must remain stationary relative to the observer who is measuring $$x$$, $$y$$, $$z$$, $$t$$. To secure this we mount our observer on the particle and then the interval-length $$s$$ will be $$t_2 - t_1$$, which is the time elapsed according to his clock.

We can use proper-time as generally equivalent to interval-length; but it must be admitted that the term is not very logical unless the track in question is a natural track. For any other track, the drawback to defining the interval-length as the time measured by a clock which follows the track, is that no clock could follow the track without violating the laws of nature. We may force it into the track by continually hitting it; but that treatment may not be good for its time-keeping qualities. The original definition by equation (1) is the more general definition.