Page:Eddington A. Space Time and Gravitation. 1920.djvu/65

dimensional world in more detail, it is necessary to return to real time, and face the difficulties of a strange geometry.

Consider a particular observer, $S$, and represent time according to his reckoning by distance up the page parallel to $OT$. One dimension of his space will be represented by horizontal distance parallel to $OX$; another will stand out at right angles from the page; and the reader must imagine the third as best he can. Fortunately it will be sufficient for us to consider only the one dimension of space $OX$ and deal with the phenomena of "line-land," i.e. we limit ourselves to motion to and fro in one straight line in space.

The two lines $U^{\prime }OU$, $V^{\prime }OV$, at 45° to the axes, represent the tracks of points which progress 1 unit horizontally (in space) for 1 unit vertically (in time); thus they represent points moving with unit velocity. We have chosen the velocity of light as unit velocity; hence $U^{\prime }OU$, $V^{\prime }OV$ will be the tracks of pulses of light in opposite directions along the straight line.

Any event $P$ within the sector $UOV$ is indubitably after the event $O$, whatever system of time-reckoning is adopted. For it would be possible for a material particle to travel from $O$ to $P$, the necessary velocity being less than that of light; and no