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

100 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, 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 how