, will represent the space and time position of an event occurring on the embankment 186,000 miles to the right of the origin
and one second after the time zero ([Fig. IX]).
We see, then, that in our space and time graph, a point traced on our sheet of paper represents not merely a position in space along the embankment, but also an instant in time. For this reason such graph-points are known as point-events. Thus a point-event constitutes the graphical representation of an instantaneous event occurring anywhere and at any time along the embankment. The position of the point-event with respect to our space and time axes will then define without ambiguity the spatio-temporal position of the physical event with respect to the embankment and to the time zero, provided the units of measurement in the graph have been specified.
And now let us consider the representation of events that last and are not merely instantaneous. Here let us note that the existence of a body, say a stone on the embankment, constitutes an event, since the position of the stone can be defined in space and in time. But the stone endures: its existence is not merely momentary; hence its permanency is given by a continuous succession of point-events forming a continuous line. This line giving the successive positions of the stone both in space and in time is called a world-line. For the stone to possess a world-line, it is not necessary that it should be in motion along the embankment; it may just as well remain at rest. The sole difference will be that if the stone is at rest, its world-line will be a vertical, whereas if it is in motion along the embankment, its world-line will be slanting, since in this case the spatial position of the stone will vary as time passes by. If the speed of the stone along the embankment is constant, its world-line will be straight, whereas if the motion is uneven or accelerated, the world-line will be more or less curved. Assuming the motion to be uniform, the greater the speed of the body, the more will its world-line slant away from the vertical and tend to become horizontal.
Fig X
Of course, as can easily be understood, the slants of the world-lines in the graph (aside from exceptional ones such as that of a body at rest) will be influenced by our choice of units. With the particular units we have chosen, the world-lines of bodies moving along the embankment at a speed of 186,000 miles a second will possess a slant of 45° with respect to both space and time axes. In other words, the world-lines of such bodies, hence also of rays of light, will be inclined equally to our space and time axes. Thus any straight line inclined in this way, either to the right or to the left ([Fig. X]), will represent the world-line of a body moving with respect to the embankment with the speed of light, either to the right or to the left. The reason we selected our units of space and time as we did, was precisely in order to confer this symmetrical position of the world-lines of light, on account of the important rôle which the velocity of light plays in the theory of relativity. We may consider still another case, that of a body moving with infinite speed along the embankment (assuming, of course, that the existence of such a motion is physically possible). The body will obviously be everywhere along the embankment at the same instant of time; hence its world-line will be a horizontal. Thus the space axis is itself the world-line of a body moving along the embankment with infinite speed at the instant zero. Conversely, the time axis is the world-line of a body remaining motionless at the origin. Again, we may say that the space axis represents the totality of events occurring on the embankment at the instant zero, hence simultaneous with one another and with the instant zero. Likewise, any horizontal represents the totality of events occurring on the embankment at some given time, the height of this horizontal above or below the space axis defining the time.
Thus far, we have been considering happenings with reference to an observer at rest on the embankment, and everything we have said applies in an identical way to classical science and to relativity. It will only be when observers in relative motion are considered that differences in our graphical representation will arise.
