Page:Eddington A. Space Time and Gravitation. 1920.djvu/99

The space-time already discussed at length in Chapter iii corresponded to the formula (2), p. 75, $ds^{2}=-dx^{2}-dy^{2}-dz^{2}+dt^{2}$. Here ($x$, $y$, $z$, $t$) are the conventional symbols for ($x_{1}$, $x_{2}$, $x_{3}$, $x_{4}$) when this special mesh-system is used, viz. rectangular coordinates and time. Comparing with (3) the potentials have the special values ${\begin{matrix}-1&0&0&0\\&-1&0&0\\&&-1&0\\&&&+1\end{matrix}}$ These are called the "Galilean values." If the potentials have these values everywhere, space-time may be called "flat," because the geometry is that of a plane surface drawn in Euclidean space of five dimensions. Recollecting what we found for two dimensions, we shall realise that a quite different set of values of the potentials may also belong to flat space-time, because the meshes may be drawn in different ways. We must clearly understand that

(1) The only way of discovering what kind of space-time is being dealt with is from the values of the potentials, which are determined practically by measurements of intervals,

(2) Different values of the potentials do not necessarily indicate different kinds of space-time,

(3) There is some complicated mathematical property common to all values of the potentials which belong to the same space-time, which is not shared by those which belong to a different kind of space-time. This property is expressed by a set of differential equations.

It can now be deduced that the space-time in which we live is not quite flat. If it were, a mesh-system could be drawn for which the $g$'s have the Galilean values, and the geometry with respect to these partitions of space and time would be that discussed in Chapter iii. For that geometry the geodesics, giving the natural tracks of particles, are straight lines.

Thus in flat space-time the law of motion is that (with suitably chosen coordinates) every particle moves uniformly in a straight line except when it is disturbed by the impacts of