THE theory of relativity appeals to what is known as the space-time graphical representation of Minkowski, but aside from certain peculiarities which the relativity theory entails, the general method of graphical representation in space and in time was known to classical science. Indeed, the graph traced by a thermometer needle is an illustration of this method. In it we have a graphic description of the variations in height of the mercury as time passes by.
The essence of these space and time graphs is to select a frame of reference, then to represent the successive positions of a body moving through this frame, in terms of its successive space and time co-ordinates. As a simple case let us consider a railroad embankment which will serve as our frame of reference. We shall restrict ourselves to considering the graphical space and time representation of events occurring on the surface of the embankment; not above it or beneath it. In other words, the space we shall be dealing with will for all practical purposes be reduced to one line, hence to one dimension. We then select a fixed point (any one at all) on the embankment and call it our origin
. Thanks to this choice of an origin, the spatial position of any event occurring on the embankment may be specified by a number. Thus the position of an event occurring two units of distance to the right or to the left of the origin will be given by the number +2 or -2, and an event occurring at the origin itself will have zero for its number.
In order to represent these results on a sheet of paper, we shall draw a straight line, say a horizontal, called a space axis; this will represent the embankment. On this space axis we mark a point
which will represent our origin on the embankment. Then, in order to represent on our paper the positions of events occurring at, say, one mile, two miles, three miles, etc., to the right of the origin on the embankment, we mark off points along our space axis at one, two and three units of distance from the point
. Obviously, we cannot manipulate a sheet of paper miles in length; hence we agree to represent a distance of one mile along the embankment by a length of one foot or one inch or one centimetre along the space axis. It matters not what unit we select so long as, once specified, it is maintained consistently throughout. As the reader can understand, the procedure is exactly the same as that followed in the plan of a city.
Thus far, our graph reduces to a space graph of the points situated along the embankment. But we have now to introduce time. Two events may happen at the same point of the embankment, but at widely different times, and our graph in its present form offers us no means of differentiating graphically between the occurrence of the two events. Accordingly, we shall agree to represent such differences in time on our sheet of paper by placing our representative points of the events at varying heights above or below the space axis. If, then, we assume that all points on our space axis represent the space and time positions of events occurring on the embankment at a time zero or at noon, it will follow that all points above or below the space axis will represent events occurring on the embankment either after or before the time zero. This is equivalent to considering a vertical axis called a time axis, along which durations, hence instants, of time will be measured. Of course, just as in the case of distances, we must agree on some unit of length in our graph, in order to represent one second in time. We may choose this unit as we please; we may, for example, represent one second by one foot, or by one inch, along the vertical