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

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 $(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}=(t_{2}-t_{1})^{2}$, 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.