Page:Eddington A. Space Time and Gravitation. 1920.djvu/115

Yet it is on the small corresponding difference in $\gamma$ that the whole of the phenomena of gravitation depend.

The coefficient $\gamma$ appears twice in the formula, and so modifies the flatness of space-time in two ways. But as a rule these two ways are by no means equally important. Its appearance as a coefficient of $dt^{2}$ produces much the most striking effects. Suppose that it is wished to measure the interval between two events in the history of a planet. If the events are, say 1 second apart in time, $dt$ = 1 second = 300,000 kilometres. Thus $dt^{2}$ = 90,000,000,000 sq. km. Now no planet moves more than 50 kilometres in a second, so that the change $dr$ associated with the lapse of 1 second in the history of the planet will not be more than 50 km. Thus $dr^{2}$ is not more than 2500 sq. km. Evidently the small term $2m/r$ has a much greater chance of making an impression where it is multiplied by $dt^{2}$ than where it is multiplied by $dr^{2}$.

Accordingly as a first approximation, we ignore the coefficient of $dr^{2}$, and consider only the meaning of $ds^{2}=-dr^{2}-r^{2}\,d\theta ^{2}+(1-2m/r)\,dt^{2}$ .........(7). We shall now show that particles situated in this kind of space-time will appear to be under the influence of an attractive force directed towards the origin.

Let us consider the problem of mapping a small portion of this kind of world on a plane.

It is first necessary to define carefully the distinction which is here drawn between a "picture" and a "map." If we are given the latitudes and longitudes of a number of places on the earth, we can make a picture by taking latitude and longitude as vertical and horizontal distances, so that the lines of latitude and longitude form a mesh-system of squares; but that does not give a true map. In an ordinary map of Europe the lines of longitude run obliquely and the lines of latitude are curved. Why is this? Because the map aims at showing as accurately as possible all distances in their true proportions^[1]. Distance is the important thing which it is desired to represent correctly. In four dimensions interval is the analogue of distance, and a map of the four-dimensional world will aim at showing all the

1. This is usually the object, though maps are sometimes made for a different purpose, e.g. Mercator's Chart.