Page:Eddington A. Space Time and Gravitation. 1920.djvu/116

intervals in their correct proportions. Our natural picture of space-time takes $r$ and $t$ as horizontal and vertical distances, e.g. when we plot the graph of the motion of a particle; but in a true map, representing the intervals in their proper proportions, the $r$ and $t$ lines run obliquely or in curves across the map.

The instructions for drawing latitude and longitude lines ($\beta$, $\lambda$) on a map, are summed up in the formula for $ds$, p. 79, $ds^{2}=d\beta ^{2}+\cos ^{2}\beta \,d\lambda ^{2}$

and similarly the instructions for drawing the $r$ and $t$ lines are given by the formula (7).

The map is shown in Fig. 14. It is not difficult to see why the $t$-lines converge to the left of the diagram. The factor $1-2m/r$ decreases towards the left where $r$ is small; and consequently any change of $t$ corresponds to a shorter interval, and must be represented in the map by a shorter distance on the left. It is less easy to see why the $r$-lines take the courses shown; by analogy with latitude and longitude we might expect them to be curved the other way. But we discussed in Chapter iii how