Page:Eddington A. Space Time and Gravitation. 1920.djvu/117

the slope of the time-direction is connected with the slope of the space-direction; and it will be seen that the map gives approximately diamond-shaped partitions of the kind represented in Fig. 6^[1].

Like all maps of curved surfaces, the diagram is only accurate in the limit when the area covered is very small.

It is important to understand clearly the meaning of this map. When we speak in the ordinary way of distance from the sun and the time at a point in the solar system, we mean the two variables $r$ and $t$. These are not the result of any precise measures with scales and clocks made at a point, but are mathematical variables most appropriate for describing the whole solar system.

They represent a compromise, because it is necessary to deal with a region too large for accurate representation on a plane map. We should naturally picture them as rectangular coordinates partitioning space-time into square meshes, as in Fig. 15; but such a picture is not a true map, because it does not represent in their true proportions the intervals between the various points in the picture. It is not possible to draw any map of the whole curved region without distortion; but a small enough portion can be represented without distortion if the partitions of equal $r$ and $t$ are drawn as in Fig. 14. To get back

1. The substitution $x=r+{\tfrac {1}{2}}t^{2}m/r$, $y=t(1-m/r)$, gives $ds*=-dx\,z+dy*$, if squares of $m$ are negligible. The map is drawn with $x$ and $y$ as rectangular coordinates.